' 


"75 


gale  bicentennial  publications 

VECTOR    ANALYSIS 


gale  'Bicentennial  publications 

With  the  approval  of  the  President  and  Fellows 
of  Tale  University,  a  series  of  volumes  has  been 
prepared  by  a  number  of  the  Professors  and  In- 
structors, to  be  issued  in  connection  with  the 
Bicentennial  Anniversary,  as  a  partial  indica- 
tion of  the  character  of  the  studies  in  which  the 
University  teachers  are  engaged. 

This   series    of  volumes    is    respectfully   dedicated  to 

of  tijc 


VECTOR  ANALYSIS 


A   TEXT-BOOK   FOR   THE   USE  OF  STUDENTS 
OF   MATHEMATICS   AND   PHYSICS 


FOUNDED  UPON  THE  LECTURES  OF 

J.  WILLARD    GIBBS,  PH.D.,  LL.D. 

Professor  of  Mathematical  Physics  in  Yale  University 


BY 

EDWIN   BIDWELL   WILSON,  PH.D. 

Instructor  in  Mathematics  in  Yale  University 


NEW  YORK :    CHARLES  SCRIBNER'S   SONS 

LONDON:   EDWARD  ARNOLD 

1901 


Copyright,  1901, 
BY  YALE   UNIVERSITY. 


Published,  December,  1901. 


UNIVERSITY    PRESS    •    JOHN    WILSON 
AND     SON     •    CAMBRIDGE,     U.S.A. 


PKEFACE  BY  PROFESSOR  GIBBS 

SINCE  the  printing  of  a  short  pamphlet  on  the  Elements  of 
Vector  Analysis  in  the  years  1881—84,  —  never  published,  but 
somewhat  widely  circulated  among  those  who  were  known  to 
be  interested  in  the  subject,  —  the  desire  has  been  expressed 
in  more  than  one  quarter,  that  the  substance  of  that  trea- 
tise, perhaps  in  fuller  form,  should  be  made  accessible  to 
the  public. 

As,  however,  the  years  passed  without  my  finding  the 
leisure  to  meet  this  want,  which  seemed  a  real  one,  I  was 
very  glad  to  have  one  of  the  hearers  of  my  course  on  Vector 
Analysis  in  the  year  1899-1900  undertake  the  preparation  of 
a  text-book  on  the  subject. 

I  have  not  desired  that  Dr.  Wilson  should  aim  simply 
at  the  reproduction  of  my  lectures,  but  rather  that  he  should 
use  his  own  judgment  in  all  respects  for  the  production  of  a 
text-book  in  which  the  subject  should  be  so  illustrated  by  an 
adequate  number  of  examples  as  to  meet  the  wants  of  stu- 
dents of  geometry  and  physics. 

J.  WILLARD  GIBBS. 
YALE  UNIVERSITY,  September,  1901. 


G47442 

BHGINKEKING  LIBRARY 


GENERAL  PREFACE 

WHEN  I  undertook  to  adapt  the  lectures  of  Professor  Gibbs 
on  VECTOR  ANALYSIS  for  publication  in  the  Yale  Bicenten- 
nial Series,  Professor  Gibbs  himself  was  already  so  fully 
engaged  upon  his  work  to  appear  in  the  same  series,  Elementary 
Principles  in  Statistical  Mechanics,  that  it  was  understood  no 
material  assistance  in  the  composition  of  this  book  could  be 
expected  from  him.  For  this  reason  he  wished  me  to  feel 
entirely  free  to  use  my  own  discretion  alike  in  the  selection 
of  the  topics  to  be  treated  and  in  the  mode  of  treatment. 
It  has  been  my  endeavor  to  use  the  freedom  thus  granted 
only  in  so  far  as  was  necessary  for  presenting  his  method  in 
text-book  form. 

By  far  the  greater  part  of  the  material  used  in  the  follow- 
ing pages  has  been  taken  from  the  course  of  lectures  on 
Vector  Analysis  delivered  annually  at  the  University  by 
Professor  Gibbs.  Some  use,  however,  has  been  made  of  the 
chapters  on  Vector  Analysis  in  Mr.  Oliver  Heaviside's  Elec~ 
tromagnetic  Theory  (Electrician  Series,  1893)  and  in  Professor 
Foppl's  lectures  on  Die  MaxwelVsche  Theorie  der  Electricitdt 
(Teubner,  1894).  My  previous  study  of  Quaternions  has 
also  been  of  great  assistance. 

The  material  thus  obtained  has  been  arranged  in  the  way 
which  seems  best  suited  to  easy  mastery  of  the  subject. 
Those  Arts,  which  it  seemed  best  to  incorporate  in  the 
text  but  which  for  various  reasons  may  well  be  omitted  at 
the  first  reading  have  been  marked  with  an  asterisk  (*).  Nu- 
merous illustrative  examples  have  been  drawn  from  geometry, 
mechanics,  and  physics.  Indeed,  a  large  part  of  the  text  has 
to  do  with  applications  of  the  method.  These  applications 
have  not  been  set  apart  in  chapters  by  themselves,  but  have 


x  GENERAL  PREFACE 

been  distributed  throughout  the  body  of  the  book  as  fast  as 
the  analysis  has  been  developed  sufficiently  for  their  adequate 
treatment.  It  is  hoped  that  by  this  means  the  reader  may  be 
better  enabled  to  make  practical  use  of  the  book.  Great  care 
has  been  taken  in  avoiding  the  introduction  of  unnecessary 
ideas,  and  in  so  illustrating  each  idea  that  is  introduced  as 
to  make  its  necessity  evident  and  its  meaning  easy  to  grasp. 
Thus  the  book  is  not  intended  as  a  complete  exposition  of 
the  theory  of  Vector  Analysis,  but  as  a  text-book  from  which 
so  much  of  the  subject  as  may  be  required  for  practical  appli- 
cations may  be  learned.  Hence  a  summary,  including  a  list 
of  the  more  important  formulae,  and  a  number  of  exercises, 
have  been  placed  at  the  end  of  each  chapter,  and  many  less 
essential  points  in  the  text  have  been  indicated  rather  than 
fully  worked  out,  in  the  hope  that  the  reader  will  supply  the 
details.  The  summary  may  be  found  useful  in  reviews  and 
for  reference. 

The  subject  of  Vector  Analysis  naturally  divides  itself  into 
three  distinct  parts.  First,  that  which  concerns  addition  and 
the  scalar  and  vector  products  of  vectors.  Second,  that  which 
concerns  the  differential  and  integral  calculus  in  its  relations 
to  scalar  and  vector  functions.  Third,  that  which  contains 
the  theory  of  the  linear  vector  function.  The  first  part  is 
a  necessary  introduction  to  both  other  parts.  The  second 
and  third  are  mutually  independent.  Either  may  be  taken 
up  first.  For  practical  purposes  in  mathematical  physics  the 
second  must  be  regarded  as  more  elementary  than  the  third. 
But  a  student  not  primarily  interested  in  physics  would  nat- 
urally pass  from  the  first  part  to  the  third,  which  he  would 
probably  find  more  attractive  and  easy  than  the  second. 

Following  this  division  of  the  subject,  the  main  body  of 
the  book  is  divided  into  six  chapters  of  which  two  deal  with 
each  of  the  three  parts  in  the  order  named.  Chapters  I.  and 
II.  treat  of  addition,  subtraction,  scalar  multiplication,  and 
the  scalar  and  vector  products  of  vectors.  The  exposition 
has  been  made  quite  elementary.  It  can  readily  be  under- 
stood by  and  is  especially  suited  for  such  readers  as  have  a 
knowledge  of  only  the  elements  of  Trigonometry  and  Ana- 


GENERAL  PREFACE  xi 

lytic  Geometry.  Those  who  are  well  versed  in  Quaternions 
or  allied  subjects  may  perhaps  need  to  read  only  the  sum- 
maries. Chapters  III.  and  IV.  contain  the  treatment  of 
those  topics  in  Vector  Analysis  which,  though  of  less  value 
to  the  students  of  pure  mathematics,  are  of  the  utmost  impor- 
tance to  students  of  physics.  Chapters  V.  and  VI.  deal  with 
the  linear  vector  function.  To  students  of  physics  the  linear 
vector  function  is  of  particular  importance  in  the  mathemati- 
cal treatment  of  phenomena  connected  with  non-isotropic 
media ;  and  to  the  student  of  pure  mathematics  this  part  of 
the  book  will  probably  be  the  most  interesting  of  all,  owing 
to  the  fact  that  it  leads  to  Multiple  Algebra  or  the  Theory 
of  Matrices.  A  concluding  chapter,  VII.,  which  contains  the 
development  of  certain  higher  parts  of  the  theory,  a  number 
of  applications,  and  a  short  sketch  of  imaginary  or  complex 
vectors,  has  been  added. 

In  the  treatment  of  the  integral  calculus,  Chapter  IV., 
questions  of  mathematical  rigor  arise.  Although  modern 
theorists  are  devoting  much  time  and  thought  to  rigor,  and 
although  they  will  doubtless  criticise  this  portion  of  the  book 
adversely,  it  has  been  deemed  best  to  give  but  little  attention 
to  the  discussion  of  this  subject.  And  the  more  so  for  the 
reason  that  whatever  system  of  notation  be  employed  ques- 
tions of  rigor  are  indissolubly  associated  with  the  calculus 
and  occasion  no  new  difficulty  to  the  student  of  Vector 
Analysis,  who  must  first  learn  what  the  facts  are  and  may 
postpone  until  later  the  detailed  consideration  of  the  restric- 
tions that  are  put  upon  those  facts. 

Notwithstanding  the  efforts  which  have  been  made  during 
more  than  half  a  century  to  introduce  Quaternions  into 
physics  the  fact  remains  that  they  have  not  found  wide  favor. 
On  the  other  hand  there  has  been  a  growing  tendency  espe- 
cially in  the  last  decade  toward  the  adoption  of  some  form  of 
Vector  Analysis.  The  works  of  Heaviside  and  Foppl  re- 
ferred to  before  may  be  cited  in  evidence.  As  yet  however 
no  system  of  Vector  Analysis  which  makes  any  claim  to 
completeness  has  been  published.  In  fact  Heaviside  says : 
"I  am  in  hopes  that  the  chapter  which  I  now  finish  may 


xii  GENERAL  PREFACE 

serve  as  a  stopgap  till  regular  vectorial  treatises  come  to  be 
written  suitable  for  physicists,  based  upon  the  vectorial  treat- 
ment of  vectors  "  (Electromagnetic  Theory,  Vol.  I.,  p.  305). 
Elsewhere  in  the  same  chapter  Heaviside  has  set  forth  the 
claims  of  vector  analysis  as  against  Quaternions,  and  others 
have  expressed  similar  views. 

The  keynote,  then,  to  any  system  of  vector  analysis  must 
be  its  practical  utility.  This,  I  feel  confident,  was  Professor 
Gibbs's  point  of  view  in  building  up  his  system.  He  uses  it 
entirely  in  his  courses  on  Electricity  and  Magnetism  and  on 
Electromagnetic  Theory  of  Light.  In  writing  this  book  I 
have  tried  to  present  the  subject  from  this  practical  stand- 
point, and  keep  clearly  before  the  reader's  mind  the  ques- 
tions: What  combinations  or  functions  of  vectors  occur  in 
physics  and  geometry  ?  And  how  may  these  be  represented 
symbolically  in  the  way  best  suited  to  facile  analytic  manip- 
ulation ?  The  treatment  of  these  questions  in  modern  books 
on  physics  has  been  too  much  confined  to  the  addition  and 
subtraction  of  vectors.  This  is  scarcely  enough.  It  has 
been  the  aim  here  to  give  also  an  exposition  of  scalar  and 
vector  products,  of  the  operator  v»  °f  divergence  and  curl 
which  have  gained  such  universal  recognition  since  the  ap- 
pearance of  Maxwell's  Treatise  on  Electricity  and  Magnetism, 
of  slope,  potential,  linear  vector  function,  etc.,  such  as  shall 
be  adequate  for  the  needs  of  students  of  physics  at  the 
present  day  and  adapted  to  them. 

It  has  been  asserted  by  some  that  Quaternions,  Vector 
Analysis,  and  all  such  algebras  are  of  little  value  for  investi- 
gating questions  in  mathematical  physics.  Whether  this 
assertion  shall  prove  true  or  not,  one  may  still  maintain  that 
vectors  are  to  mathematical  physics  what  invariants  are  to 
geometry.  As  every  geometer  must  be  thoroughly  conver- 
sant with  the  ideas  of  invariants,  so  every  student  of  physics 
should  be  able  to  think  in  terms  of  vectors.  And  there  is 
no  way  in  which  he,  especially  at  the  beginning  of  his  sci- 
entific studies,  can  come  to  so  true  an  appreciation  of  the 
importance  of  vectors  and  of  the  ideas  connected  with  them 
as  by  working  in  Vector  Analysis  and  dealing  directly  with 


GENERAL  PREFACE  xiii 

the  vectors  themselves.  To  those  that  hold  these  views  the 
success  of  Professor  Fb'ppl's  Vorlesungen  uber  Technische 
Mechanik  (four  volumes,  Teubner,  1897-1900,  already  in  a 
second  edition),  in  which  the  theory  of  mechanics  is  devel- 
oped by  means  of  a  vector  analysis,  can  be  but  an  encour- 
aging sign. 

I  take  pleasure  in  thanking  my  colleagues,  Dr.  M.  B.  Porter 
and  Prof.  H.  A.  Bumstead,  for  assisting  me  with  the  manu- 
script. The  good  services  of  the  latter  have  been  particularly 
valuable  in  arranging  Chapters  III.  and  IV.  in  their  present 
form  and  in  suggesting  many  of  the  illustrations  used  in  the 
work.  I  am  also  under  obligations  to  my  father,  Mr.  Edwin 
H.  Wilson,  for  help  in  connection  both  with  the  proofs  and 
the  manuscript.  Finally,  I  wish  to  express  my  deep  indebt- 
edness to  Professor  Gibbs.  For  although  he  has  been  so 
preoccupied  as  to  be  unable  to  read  either  manuscript  or 
proof,  he  has  always  been  ready  to  talk  matters  over  with 
me,  and  it  is  he  who  has  furnished  me  with  inspiration  suf- 
ficient to  cany  through  the  work. 

EDWIN  BIDWELL  WILSON. 

YALE  UNIVERSITY,  October,  1901. 


TABLE   OF   CONTENTS 


PAGE 
PREFACE  BY  PROFESSOR  GIBBS vii 

GENERAL  PREFACE ix 


CHAPTER  I 

ADDITION  AND   SCALAR  MULTIPLICATION 
ARTS. 

1-3     SCALARS  AND  VECTORS 1 

4  EQUAL  AND  NULL  VECTORS 4 

5  THE  POINT  OF  VIEW  OF  THIS  CHAPTER 6 

6-7     SCALAR  MULTIPLICATION.     THE  NEGATIVE  SIGN  ....  7 

8-10    ADDITION.     THE  PARALLELOGRAM  LAW 8 

11  SUBTRACTION 11 

12  LAWS  GOVERNING  THE  FOREGOING  OPERATIONS     ....  12 
13-16     COMPONENTS  OF  VECTORS.     VECTOR  EQUATIONS  ....  14 

17      THE    THREE    UNIT    VECTORS   1,  j,  k 18 

18-19     APPLICATIONS  TO  SUNDRY  PROBLEMS  IN  GEOMETRY.     .     .  21 

20-22     VECTOR  RELATIONS  INDEPENDENT  OF  THE  ORIGIN    ...  27 

23-24     CENTERS  OF  GRAVITY.     BARYCENTRIC  COORDINATES     .     .  39 

25    THE  USE  OF  VECTORS  TO  DENOTE  AREAS 46 

SUMMARY  OF  CHAPTER  i 51 

EXERCISES  ON  CHAPTER  i  52 


CHAPTER  II 
DIRECT  AND   SKEW  PRODUCTS  OF  VECTORS 

27-28     THE  DIRECT,  SCALAR,  OR  DOT  PRODUCT  OF  TWO  VECTORS  55 

29-30     THE  DISTRIBUTIVE  LAW  AND  APPLICATIONS 58 

31-33     THE  SKEW,  VECTOR,  OR  CROSS  PRODUCT  OF  TWO  VECTORS  60 

34-35     THE  DISTRIBUTIVE  LAW  AND  APPLICATIONS 63 

36    THE  TRIPLE  PRODUCT  A*  B  C        67 


XVI 


CONTENTS 


ARTS. 

37-38    THE  SCALAR  TRIPLE  PRODUCT  A*  B  X  COR  [ABC]      •     .  68 

39-40    THE  VECTOR  TRIPLE  PRODUCT  A  X  (B  X  C) 71 

41-42    PRODUCTS  OF   MORE  THAN  THREE  VECTORS  WITH   APPLI- 
CATIONS TO  TRIGONOMETRY 75 

43-45    RECIPROCAL  SYSTEMS  OF  THREE  VECTORS 81 

46-47    SOLUTION  OF  SCALAR  AND  VECTOR  EQUATIONS  LINEAR  IN 

AN  UNKNOWN  VECTOR       87 

48-50     SYSTEMS  OF  FORCES  ACTING  ON  A  RIGID  BODY  ....  92 

51  KINEMATICS  OF  A  RIGID  BODY 97 

52  CONDITIONS  FOR  EQUILIBRIUM  OF  A  RIGID  BODY    .     .     .  101 

53  RELATIONS_JBETWEEN    TWO   RIGHT-HANDED   SYSTEMS    OF 

TIIKKK  I'KKI'I.M>I<TLAU  T  N  IT  VKCTOKS 104 

54  PROBLEMS  IN  GEOMETRY.     PLANAR  COORDINATES  .     .     .  106 

SUMMARY  OF  CHAPTER  n 109 

EXERCISES  ON  CHAPTER  n 113 


CHAPTER  HI 
THE  DIFFERENTIAL  CALCULUS  OF  VECTORS 

55-56     DERIVATIVES  AND  DIFFERENTIALS  OF  VECTOR  FUNCTIONS 

WITH  RESPECT  TO  A  SCALAR  VARIABLE 115 

57    CURVATURE  AND  TORSION  OF  GAUCHE  CURVES  ....  120 

58-59    KINEMATICS  OF  A  PARTICLE.     THE  HODOGRAPH     .     .     .  125 

60  THE  INSTANTANEOUS  AXIS  OF  ROTATION 131 

61  INTEGRATION  WITH  APPLICATIONS  TO  KINEMATICS  .     .     .  133 

62  SCALAR  FUNCTIONS  OF  POSITION  IN  SPACE 136 

63-67     THE  YECTORJDIFFEUENTIATING  OPERATOR  V     •     •     •     •  138 

68  THE  SCALAR  OPERATOR  A  *  'v7 147 

69  VECTOR  FUNCTIONS  OF  POSITION  IN  SPACE 149 

70  THE  DIVERGENCE  V»   AND  THE  CURL  V  X 150 

71  INTERPRETATION  OF  THE  DIVERGENCE  S7 152 

72  INTERPRETATION  OF  THE  CURL  V  X 155 

73  LAWS  OF    OPERATION   OF   V>   V  *   >   V  X 157' 

74-76     THE  PARTIAL  APPLICATION  OF  V-     EXPANSION  OF  A  VEC- 
TOR    FUNCTION    ANALOGOUS     TO     TAYLOR'S    THEOREM. 

APPLICATION  TO  HYDROMECHANICS 159 

77  THE  DIFFERENTIATING  OPERATORS  OF  THE  SECOND  ORDER  166 

78  GEOMETRIC   INTERPRETATION    OF    LAPLACE'S    OPERATOR 

V*  V  As  THE  DISPERSION 170 

SUMMARY  OF  CHAPTER  in 172 

EXERCISES  ON  CHAPTER  in  177 


CONTENTS  xvii 


CHAPTER  IV 

THE  INTEGRAL  CALCULUS  OF  VECTORS 

ARTS.  PAGE 
79-80    LINE  INTEGRALS  OF  VECTOR  FUNCTIONS  WITH  APPLICA- 
TIONS        179 

81  GAUSS'S  THEOREM 184 

82  STOKKS'S  THEOREM 187 

83  CONVERSE  OF  STOKES'S  THEOREM  WITH  APPLICATIONS     .  193 

84  TRANSFORMATIONS  OF   LINE,  SURFACE,  AND  VOLUME   IN- 

TEGRALS.    GREEN'S  THEOREM 197 

85  REMARKS  ON  MULTIPLE-VALUED  FUNCTIONS 200 

86-87     POTENTIAL.     THE  INTEGRATING  OPERATOR  "  POT  "      .     .  205 

88  COMMUTATIVE  PROPERTY  OF  POT  AND  ^7 211 

89  REMARKS  UPON  THE  FOREGOING 215 

90  THE  INTEGRATING  OPERATORS  "NEW,"  "LAP,"  "  MAX  "  222 

91  RELATIONS    BETWEEN    THE    INTEGRATING   AND    DIFFER- 

ENTIATING OPERATORS 228 

92  THE    POTENTIAL    "  POT  "    is   A   SOLUTION  OF    POISSON'S 

EQUATION 230 

93-94     SOLENOIDAL    AND    IRROTATIONAL    PARTS   OF   A  VECTOR 

FUNCTION.     CERTAIN  OPERATORS  AND  THEIR  INVERSE    .  234 

95  MUTUAL   POTENTIALS,    NEWTONIANS,  LAPLACIANS,   AND 

MAXWELLIANS 240 

96  CERTAIN  BOUNDARY  VALUE  THEOREMS       243 

SUMMARY  OF  CHAPTER  iv .  249 

EXERCISES  ON  CHAPTER  iv  .  255 


CHAPTER  V 

LINEAR  VECTOR  FUNCTIONS 

97-98     LINEAR  VECTOR  FUNCTIONS  DEFINED 260 

99     DYADICS  DEFINED 264 

100  ANY  LINEAR  VECTOR  FUNCTION  MAY  BE  REPRESENTED 

BY  A  DYADIC.     PROPERTIES  OF  DYADICS      ....     266 

101  THE  NONION  FORM  OF  A  DYADIC 269 

102  THE  DYAD  OR  INDETERMINATE  PRODUCT  OF  TWO  VEC- 

TORS IS  THE  MOST  GENERAL.   FUNCTIONAL  PROPERTY 

OF  THE  SCALAR  AND  VECTOR  PRODUCTS 271 

103-104     PRODUCTS  OF  DYADICS 276 

105-107    DEGREES  OF  NULLITY  OF  DYADICS 282 

108     THE  IDEMF ACTOR  ...  ...  288 


XV111 


CONTENTS 


ARTS.  PAGE 
109-110    RECIPROCAL  DYADICS.    POWERS  AND  ROOTS  OF  DYADICS  290 
111     CONJUGATE    DYADICS.       SELF-CONJUGATE    AND    ANTI- 
SELF-CONJUGATE  PARTS  OF  A  DYADIC 294 

112-114    ANTI-SELF-CONJUGATE    DYADICS.      THE  VECTOR  PROD- 
UCT.     QUADRANTAL   VERSORS 297 

115-116     REDUCTION  OF  DYADICS  TO  NORMAL  FORM      ....  302 

117    DOUBLE  MULTIPLICATION  OF  DYADICS 306 

118-119     THE  SECOND  AND  THIRD  OF  A  DYADIC 310 

120  CONDITIONS  FOR  DIFFERENT  DEGREES  OF  NULLITY       .  313 

121  NONION  FORM.     DETERMINANTS 315 

122  INVARIANTS  OF   A  DYADIC.      THE  HAMILTON-CAYLEY 

EQUATION 319 

SUMMARY  OF  CHAPTER  v 321 

EXERCISES  ON  CHAPTER  v 329 


CHAPTER  VI 

ROTATIONS  AND  STRAINS 

123-124    HOMOGENEOUS  STRAIN  REPRESENTED  BY  A  DYADIC      .  332 

125-126    ROTATIONS  ABOUT  A  FIXED  POINT.     VERSORS     .     .     .  334 

127  THE  VECTOR  SEMI-TANGENT  OF  VERSION     .     .     .     .     .  339 

128  BlQUADRANTAL  VERSORS  AND  THEIR  PRODUCTS  .     .     .  343 

129  CYCLIC  DYADICS 347 

130  RIGHT  TENSORS 351 

131  TONICS  AND  CYCLOTONICS 353 

132  REDUCTION  OF  DYADICS  TO  CANONICAL  FORMS,  TONICS, 

CYCLOTONICS,  SIMPLE  AND  COMPLEX  SHEARERS     .     .  356 

SUMMARY  OF  CHAPTER  vi 368 

CHAPTER  VII 

MISCELLANEOUS  APPLICATIONS 

136-142    QUADRIC  SURFACES 372 

143-146     THE  PROPAGATION  OF  LIGHT  IN  CRYSTALS       ....  392 

147-148    VARIABLE  DYADICS 403 

149-157     CURVATURE  OF  SURFACES 411 

158-162    HARMONIC  VIBRATIONS  AND  BIVECTORS 426 


YECTOft   ANALYSIS 


VECTOR   ANALYSIS 


CHAPTER   I 

ADDITION  AND   SCALAR    MULTIPLICATION 

1.]  IN  mathematics  and  especially  in  physics  two  very- 
different  kinds  of  quantity  present  themselves.  Consider,  for 
example,  mass,  time,  density,  temperature,  force,  displacement 
of  a  point,  velocity,  and  acceleration.  Of  these  quantities 
some  can  be  represented  adequately  by  a  single  number  — 
temperature,  by  degrees  on  a  thermometric  scale ;  time,  by 
3^ears,  days,  or  seconds ;  mass  and  density,  by  numerical  val- 
ues which  are  wholly  determined  when  the  unit  of  the  scale 
is  fixed.  On  the  other  hand  the  remaining  quantities  are  not 
capable  of  such  representation.  Force  to  be  sure  is  said  to  be 
of  so  many  pounds  or  grams  weight;  velocity,  of  so  many 
feet  or  centimeters  per  second.  But  in  addition  to  this  each 
-of  them  must  be  considered  as  having  direction  as  well  as 
magnitude.  A  force  points  North,  South,  East,  West,  up, 
down,  or  in  some  intermediate  direction.  The  same  is  true 
of  displacement,  velocity,  and  acceleration.  No  scale  of  num- 
bers can  represent  them  adequately.  It  can  represent  only 
their  magnitude,  not  their  direction. 

2.]  Definition  :  A  vector  is  a  quantity  which  is  considered 
as  possessing  direction  as  well  as  magnitude. 

Definition :  A  scalar  is  a  quantity  which  is  considered  as  pos- 
sessing magnitude  but  no  direction. 


VECTOR  ANALYSIS 

The  positive  and  negative  numbers  of  ordinary  algebra  are  the 
typical  scalar s.  For  this  reason  the  ordinary  algebra  is  called 
scalar  algebra  when  necessary  to  distinguish  it  from  the  vector 
algebra  or  analysis  which  is  the  subject  of  this  book. 

The  typical  vector  is  the  displacement  of  translation  in  space. 
Consider  first  a  point  P  (Fig.  1).  Let  P  be  displaced  in  a 
,  straight  line  and  take  a  new  position  P'. 
This  change  of  position  is  represented  by  the 
line  PP.  The  magnitude  of  the  displace- 
ment is  the  length  of  PP' ;  the  direction  of 
it  is  the  direction  of  the  line  PP'  from  P  to 
P'.  Next  consider  a  displacement  not  of  one, 
but  of  all  the  points  in  space.  Let  all  the 
points  move  in  straight  lines  in  the  same  direction  and  for  the 
same  distance  D.  This  is  equivalent  to  shifting  space  as  a 
rigid  body  in  that  direction  through  the  distance  D  without 
rotation.  Such  a  displacement  is  called  a  translation.  It 
possesses  direction  and  magnitude.  When  space  undergoes 
a  translation  T,  each  point  of  space  undergoes  a  displacement 
equal  to  T  in  magnitude  and  direction;  and  conversely  if 
the  displacement  PP'  which  any  one  particular  point  P  suf- 
fers in  the  translation  T  is  known,  then  that  of  any  other 
point  Q  is  also  known :  for  Q  Q'  must  be  equal  and  parallel 
toPP'. 

The  translation  T  is  represented  geometrically  or  graphically 
by  an  arrow  T  (Fig.  1)  of  which  the  magnitude  and  direction 
are  equal  to  those  of  the  translation.  The  absolute  position 
of  this  arrow  in  space  is  entirely  immaterial.  Technically  the 
arrow  is  called  a  stroke.  Its  tail  or  initial  point  is  its  origin  ; 
and  its  head  or  final  point,  its  terminus.  In  the  figure  the 
origin  is  designated  by  0  and  the  terminus  by  T.  This  geo- 
metric quantity,  a  stroke,  is  used  as  the  mathematical  symbol 
for  all  vectors,  just  as  the  ordinary  positive  and  negative  num- 
bers are  used  as  the  symbols  for  all  scalars. 


ADDITION  AND  SCALAR  MULTIPLICATION  3 

*  3.]  As  examples  of  scalar  quantities  mass,  time,  den- 
sity, and  temperature  have  been  mentioned.  Others  are  dis- 
tance, volume,  moment  of  inertia,  work,  etc.  Magnitude, 
however,  is  by  no  means  the  sole  property  of  these  quantities. 
Each  implies  something  besides  magnitude.  Each  has  its 
own  distinguishing  characteristics,  as  an  example  of  which 
its  dimensions  in  the  sense  well  known  to  physicists  may 
be  cited.  A  distance  3,  a  time  3,  a  work  3,  etc.,  are  very- 
different.  The  magnitude  3  is,  however,  a  property  common 
to  them  all  —  perhaps  the  only  one.  Of  all  scalar  quanti- 
tities  pure  number  is  the  simplest.  It  implies  nothing  but 
magnitude.  It  is  the  scalar  par  excellence  and  consequently 
it  is  used  as  the  mathematical  symbol  for  all  scalars. 

As  examples  of  vector  quantities  force,  displacement,  velo- 
city, and  acceleration  have  been  given.  Each  of  these  has 
other  characteristics  than  those  which  belong  to  a  vector  pure 
and  simple.  The  concept  of  vector  involves  two  ideas  and 
two  alone  —  magnitude  of  the  vector  and  direction  of  the 
vector.  But  force  is  more  complicated.  When  it  is  applied 
,to  a  rigid  body  the  line  in  which  it  acts  must  be  taken  into 
consideration ;  magnitude  and  direction  alone  do  not  suf- 
fice. And  in  case  it  is  applied  to  a  non-rigid  body  the  point 
of  application  of  the  force  is  as  important  as  the  magnitude  or 
direction.  Such  is  frequently  true  for  vector  quantities  other 
than  force.  Moreover  the  question  of  dimensions  is  present 
as  in  the  case  of  scalar  quantities.  The  mathematical  vector, 
the  stroke,  which  is  the  primary  object  of  consideration  in 
this  book,  abstracts  from  all  directed  quantities  their  magni- 
tude and  direction  and  nothing  but  these ;  just  as  the  mathe- 
matical scalar,  pure  number,  abstracts  the  magnitude  and 
that  alone.  Hence  one  must  be  on  his  guard  lest  from 
analogy  he  attribute  some  properties  to  the  mathematical 
vector  which  do  not  belong  to  it ;  and  he  must  be  even  more 
careful  lest  he  obtain  erroneous  results  by  considering  the 


4  VECTOR  ANALYSIS 

vector  quantities  of  physics  as  possessing  no  properties  other 
than  those  of  the  mathematical  vector.  For  example  it  would 
never  do  to  consider  force  and  its  effects  as  unaltered  by 
shifting  it  parallel  to  itself.  This  warning  may  not  be 
necessary,  yet  it  may  possibly  save  some  confusion. 

4.]  Inasmuch  as,  taken  in  its  entirety,  a  vector  or  stroke 
is  but  a  single  concept,  it  may  appropriately  be  designated  by 
one  letter.  Owing  however  to  the  fundamental  difference 
between  scalars  and  vectors,  it  is  necessary  to  distinguish 
carefully  the  one  from  the  other.  Sometimes,  as  in  mathe- 
matical physics,  the  distinction  is  furnished  by  the  physical 
interpretation.  Thus  if  n  be  the  index  of  refraction  it 
must  be  scalar;  m,  the  mass,  and  £,  the  time,  are  also 
scalars ;  but  /,  the  force,  and  a,  the  acceleration,  are 
vectors.  When,  however,  the  letters  are  regarded  merely 
as  symbols  with  no  particular  physical  significance  some 
typographical  difference  must  be  relied  upon  to  distinguish 
vectors  from  scalars.  Hence  in  this  book  Clarendon  type  is 
used  for  setting  up  vectors  and  ordinary  type  for  scalars. 
This  permits  the  use  of  the  same  letter  differently  printed 
to  represent  the  vector  and  its  scalar  magnitude.1  Thus  if 
C  be  the  electric  current  in  magnitude  and  direction,  C  may 
be  used  to  represent  the  magnitude  of  that  current ;  if  g  be 
the  vector  acceleration  due  to  gravity,  g  may  be  the  scalar 
value  of  that  acceleration ;  if  v  be  the  velocity  of  a  moving 
mass,  v  may  be  the  magnitude  of  that  velocity.  The  use  of 
Clarendons  to  denote  vectors  makes  it  possible  to  pass  from 
directed  quantities  to  their  scalar  magnitudes  by  a  mere 
change  in  the  appearance  of  a  letter  without  any  confusing 
change  in  the  letter  itself. 

Definition :  Two  vectors  are  said  to  be  equal  when  they  have 
the  same  magnitude  and  the  same  direction. 

1  This  convention,  however,  is  by  no  means  invariably  followed.  In  some 
instances  it  would  prove  just  as  undesirable  as  it  is  convenient  in  others.  It  is 
chiefly  valuable  in  the  application  of  vectors  to  physics. 


ADDITION  AND  SCALAR  MULTIPLICATION  5 

The  equality  of  two  vectors  A  and  B  is  denoted  by  the 
usual  sign  =.  Thus  A  =  B 

Evidently  a  vector  or  stroke  is  not  altered  by  shifting  it 
about  parallel  to  itself  in  space.  Hence  any  vector  A  =  PP' 
(Fig.  1)  may  be  drawn  from  any  assigned  point  0  as  origin  ; 
for  the  segment  PP'  may  be  moved  parallel  to  itself  until 
the  point  P  falls  upon  the  point  0  and  P'  upon  some  point  T. 

A  =  PP  =  OT=  T. 

In  this  way  all  vectors  in  space  may  be  replaced  by  directed 
segments  radiating  from  one  fixed  point  0.  Equal  vectors 
in  space  will  of  course  coincide,  when  placed  with  their  ter- 
mini at  the  same  point  0.  Thus  (Fig.  1)  A  =  PP't  and  B  =  Q~Q', 
both  fall  upon  T  =  ~OT. 

For  the  numerical  determination  of  a  vector  three  scalars 
are  necessary.  These  may  be  chosen  in  a  variety  of  ways. 
If  r,  $,  0  be  polar  coordinates  in  space  any  vector  r  drawn 
with  its  origin  at  the  origin  of  coordinates  may  be  represented 
by  the  three  scalars  r,  <£,  9  which  determine  the  terminus  of 

the  vector.  ,      ,    /,, 

r  ~  (r,  $,  V) . 

Or  if  z,  y,  z  be  Cartesian  coordinates  in  space  a  vector  r  may 
be  considered  as  given  by  the  differences  of  the  coordinates  x\ 
yf,  z'  of  its  terminus  and  those  #,  y,  z  of  its  origin. 

r  ~  (x1  -  x,  y'  —  y,z'  —  z). 

If  in  particular  the  origin  of  the  vector  coincide  with  the 
origin  of  coordinates,  the  vector  will  be  represented  by  the 
three  coordinates  of  its  terminus 

r~(x',  y',  z'). 

When  two  vectors  are  equal  the  three  scalars  which  repre- 
sent them  must  be  equal  respectively  each  to  each.  Hence 
one  vector  equality  implies  three  scalar  equalities. 


6  VECTOR  ANALYSIS 

Definition  :  A  vector  A  is  said  to  be  equal  to  zero  when  its 
magnitude  A  is  zero. 

Such  a  vector  A  is  called  a  null  or  zero  vector  and  is  written 
equal  to  naught  in  the  usual  manner.  Thus 

A  =  0  if  A  =  0. 

All  null  vectors  are  regarded  as  equal  to  each  other  without 
any  considerations  of  direction. 

In  fact  a  null  vector  from  a  geometrical  standpoint  would 
be  represented  by  a  linear  segment  of  length  zero  —  that  is  to 
say,  by  a  point.  It  consequently  would  have  a  wholly  inde- 
terminate direction  or,  what  amounts  to  the  same  thing,  none  at 
all.  If,  however,  it  be  regarded  as  the  limit  approached  by  a 
vector  of  finite  length,  it  might  be  considered  to  have  that 
direction  which  is  the  limit  approached  by  the  direction  of  the 
finite  vector,  when  the  length  decreases  indefinitely  and  ap- 
proaches zero  as  a  limit.  The  justification  for  disregarding 
this  direction  and  looking  upon  all  null  vectors  as  equal  is 
that  when  they  are  added  (Art.  8)  to  other  vectors  no  change 
occurs  and  when  multiplied  (Arts.  27,  31)  by  other  vectors 
the  product  is  zero. 

5.]  In  extending  to  vectors  the  fundamental  operations 
of  algebra  and  arithmetic,  namely,  addition,  subtraction,  and 
multiplication,  care  must  be  exercised  not  only  to  avoid  self- 
contradictory  definitions  but  also  to  lay  down  useful  ones. 
Both  these  ends  may  be  accomplished  most  naturally  and 
easily  by  looking  to  physics  (for  in  that  science  vectors  con- 
tinually present  themselves)  and  by  observing  how  such 
quantities  are  treated  there.  If  then  A  be  a  given  displace- 
ment, force,  or  velocity,  what  is  two,  three,  or  in  general  x 
times  A?  What,  the  negative  of  A?  And  if  B  be  another, 
what  is  the  sum  of  A  and  B  ?  That  is  to  say,  what  is  the 
equivalent  of  A  and  B  taken  together  ?  The  obvious  answers 
to  these  questions  suggest  immediately  the  desired  definitions. 


ADDITION  AND  SCALAR  MULTIPLICATION  7 

Scalar  Multiplication 

6.]  Definition:  A  vector  is  said  to  be  multiplied  by  a 
positive  scalar  when  its  magnitude  is  multiplied  by  that  scalar 
and  its  direction  is  left  unaltered. 

Thus  if  v  be  a  velocity  of  nine  knots  East  by  North,  2  J  times 
v  is  a  velocity  of  twenty-one  knots  with  the  direction  still 
East  by  North.  Or  if  f  be  the  force  exerted  upon  the  scale- 
pan  by  a  gram  weight,  1000  times  f  is  the  force  exerted  by  a 
kilogram.  The  direction  in  both  cases  is  vertically  down- 
ward. 

If  A  be  the  vector  and  x  the  scalar  the  product  of  x  and  A  is 

denoted  as  usual  by 

x  A  or  A  x. 

It  is,  however,  more  customary  to  place  the  scalar  multiplier 
before  the  multiplicand  A.  This  multiplication  by  a  scalar 
is  called  scalar  multiplication,  and  it  follows  the  associative  law 

x  (y  A)  =  (x  y)  A  =  y  (x  A) 

as  in  ordinary  algebra  and  arithmetic.  This  statement  is  im- 
mediately obvious  when  the  fact  is  taken  into  consideration 
that  scalar  multiplication  does  not  alter  direction  but  merely 
multiplies  the  length. 

Definition :  A  unit  vector  is  one  whose  magnitude  is  unity. 

Any  vector  A  may  be  looked  upon  as  the  product  of  a  unit 
vector  a  in  its  direction  by  the  positive  scalar  A,  its  magni- 
tude. 

A  =  A  a  =  a  A. 

The  unit  vector  a  may  similarly  be  written  as  the  product  of 
A  by  \/A  or  as  the  quotient  of  A  and  A. 

1  A 


8  VECTOR  ANALYSIS 

7.]  Definition :  The  negative  sign,  — ,  prefixed  to  a  vector 
reverses  its  direction  but  leaves  its  magnitude  unchanged. 

For  example  if  A  be  a  displacement  for  two  feet  to  the  right, 
-  A  is  a  displacement  for  two  feet  to  the  left.  Again  if  the 
stroke  A  B  be  A,  the  stroke  B  A,  which  is  of  the  same  length 
as  A  B  but  which  is  in  the  direction  from  B  to  A  instead  of 
from  A  to  B,  will  be  —  A.  Another  illustration  of  the  use 
of  the  negative  sign  may  be  taken  from  Newton's  third  law 
of  motion.  If  A  denote  an  "action,"  —  A  will  denote  the 
"  reaction."  The  positive  sign,  + ,  may  be  prefixed  to  a  vec- 
tor to  call  particular  attention  to  the  fact  that  the  direction 
has  not  been  reversed.  The  two  signs  +  and  —  when  used 
in  connection  with  scalar  multiplication  of  vectors  follow  the 
same  laws  of  operation  as  in  ordinary  algebra.  These  are 
symbolically 

+  +  =  +  ;    +  -  =  -;    -  +  =  -;    --  =  +  ; 
—  (ra  A)  =  ra  (—  A). 

The  interpretation  is  obvious. 

Addition  and  Subtraction 

8.]     The  addition  of  two  vectors  or  strokes  may  be  treated 
most  simply  by  regarding  them  as  defining  translations   in 
space  (Art.  2).     Let  S  be  one  vector  and  T  the  other.     Let  P 
be  a  point  of  space  (Fig.  2).     The  trans- 
lation S  carries  P  into  P'  such  that  the 
line  PP'  is  equal  to  S  in  magnitude  and 
direction.    The  transformation  T  will  then 
carry  P'  into  P"  —  the  line  P'P"  being 
parallel  to  T  and  equal  to  it  in  magnitude. 
FIG.  2.  Consequently  the  result  of  S  followed  by 

T  is  to  carry  the  point  P  into  the  point 
P".  If  now  Q  be  any  other  point  in  space,  S  will  carry  Q 
into  Q'  such  that  QQ1  =  S  and  T  will  then  carry  Q'  into  Q" 


ADDITION  AND  SCALAR  MULTIPLICATION  9 


such  that  Q'  Q"  =  T.  Thus  S  followed  by  T  carries  Q  into  Q". 
Moreover,  the  triangle  Q  Q'  Q"  is  equal  to  PP'P".  For 
the  two  sides  Q  Q'  and  Q'  Q",  being  equal  and  parallel  to  S 
and  T  respectively,  must  be  likewise  parallel  to  P  P'  and 
P' P"  respectively  which  are  also  parallel  to  S  and  T.  Hence 
the  third  sides  of  the  triangles  must  be  equal  and  parallel. 

That  is 

Q  Q"  is  equal  and  parallel  to  PP". 

As  Q  is  any  point  in  space  this  is  equivalent  to  saying  that 
by  means  of  S  followed  by  T  all  points  of  space  are  displaced 
the  same  amount  and  in  the  same  direction.  This  displace- 
ment is  therefore  a  translation.  Consequently  the  two 
translations  S  and  T  are  equivalent  to  a  single  translation  B,. 
Moreover 


if  S  =  PP'  and  T  =  P'  P",  then  E  =  PP". 

The  stroke  B,  is  called  the  resultant  or  sum  of  the  two 
strokes  S  and  T  to  which  it  is  equivalent.  This  sum  is  de- 
noted in  the  usual  manner  by 

R  =  S  +  T. 

From  analogy  with  the  sum  or  resultant  of  two  translations 
the  following  definition  for  the  addition  of  any  two  vectors  is 
laid  down. 

Definition :  The  sum  or  resultant  of  two  vectors  is  found 
by  placing  the  origin  of  the  second  upon  the  terminus  of  the 
first  and  drawing  the  vector  from  the  origin  of  the  first  to  the 
terminus  of  the  second. 

9.]  Theorem.  The  order  in  which  two  vectors  S  and  T  are 
added  does  not  affect  the  sum. 

S  followed  by  T  gives  precisely  the  same  result  as  T  followed 
by  S.  For  let  S  carry  P  into  P'  (Fig.  3) ;  and  T,  P1  into  P". 
S  +  T  then  carries  P  into  P".  Suppose  now  that  T  carries  P 
into  P'".  The  line  PP'"  is  equal  and  parallel  to  P'P".  Con- 


10  VECTOR  ANALYSIS 

sequently  the  points  P,  P',  P",  and  P'"  lie  at  the  vertices  of 

a  parallelogram.  Hence 
pin  pn  js  equai  an(j  par_ 

allel  to  PP'.  Hence  S 
carries  P'"  into  P".  T  fol- 
lowed by  S  therefore  car- 
ries P  into  P"  through  P', 
whereas  S  followed  by  T 
carries  P  into  P"  through 
P'".  The  final  result  is  in 

either  case  the  same.     This  may  be  designated  symbolically 

by  writing 

It  is  to  be  noticed  that  S  =  PP'  and  T  =  P  P'"  are  the  two  sides 
of  the  parallelogram  PP'  P"  P'"  which  have  the  point  P  as 
common  origin ;  and  that  R  =  PP"  is  the  diagonal  drawn 
through  P.  This  leads  to  another  very  common  way  of 
stating  the  definition  of  the  sum  of  two  vectors. 

If  two  vectors  be  drawn  from  the  same  origin  and  a  parallelo- 
gram be  constructed  upon  them  as  sides,  their  sum  will  be  that 
\  diagonal  which  passes  through  their  common  origin. 

This  is  the  well-known  "  parallelogram  law  "  according  to 
which  the  physical  vector  quantities  force,  acceleration,  veloc- 
ity, and  angular  velocity  are  compounded.  It  is  important  to 
note  that  in  case  the  vectors  lie  along  the  same  line  vector 
addition  becomes  equivalent  to  algebraic  scalar  addition.  The 
lengths  of  the  two  vectors  to  be  added  are  added  if  the  vectors 
have  the  same  direction ;  but  subtracted  if  they  have  oppo- 
site directions.  In  either  case  the  sum  has  the  same  direction 
as  that  of  the  greater  vector. 

10.]  After  the  definition  of  the  sum  of  two  vectors  has 
been  laid  down,  the  sum  of  several  may  be  found  by  adding 
together  the  first  two,  to  this  sum  the  third,  to  this  the  fourth, 
and  so  on  until  all  the  vectors  have  been  combined  into  a  sin- 


ADDITION  AND  SCALAR  MULTIPLICATION         11 

gle  one.  The  final  result  is  the  same  as  that  obtained  by  placing 
the  origin  of  each  succeeding  vector  upon  the  terminus  of  the 
preceding  one  and  then  drawing  at  once  the  vector  from 
the  origin  of  the  first  to  the  terminus  of  the  last.  In  case 
these  two  points  coincide  the  vectors  form  a  closed  polygon 
and  their  sum  is  zero.  Interpreted  geometrically  this  states 
that  if  a  number  of  displacements  R,  S,  T  •  •  •  are  such  that  the 
strokes  R,  S,  T  •  •  •  form  the  sides  of  a  closed  polygon  taken  in 
order,  then  the  effect  of  carrying  out  the  displacements  is  nil. 
Each  point  of  space  is  brought  back  to  its  starting  point.  In- 
terpreted in  mechanics  it  states  that  if  any  number  of  forces 
act  at  a  point  and  if  they  form  the  sides  of  a  closed  polygon 
taken  in  order,  then  the  resultant  force  is  zero  and  the  point 
is  in  equilibrium  under  the  action  of  the  forces. 

The  order  of  sequence  of  the  vectors  in  a  sum  is  of  no  con- 
sequence. This  may  be  shown  by  proving  that  any  two  adja- 
cent vectors  may  be  interchanged  without  affecting  the  result. 

To  show 

A+B+C+D+E=A+B+D+C+R 

Let      A  =  ~0~A,  B  =  Zi?,  C  =  ~BC,  D  =  (TlT,  E  =  WE. 
Then  Ofi  =  A  +  B  +  C  +  D  +  E. 


Let  now  B  C'  =  D.  Then  C'  B  C  D  is  a  parallelogram  and 
consequently  C'  D  =  C.  Hence 

0~E  =  A  +  B  +  D  +  C  +  E, 

which  proves  the  statement.  Since  any  two  adjacent  vectors 
may  be  interchanged,  and  since  the  sum  may  be  arranged  in 
any  order  by  successive  interchanges  of  adjacent  vectors,  the 
order  in  which  the  vectors  occur  in  the  sum  is  immaterial. 

11.]     Definition:  A  vector  is  said  to  be  subtracted  when  it 
is  added  after  reversal  of  direction.     Symbolically, 

A  -  B  =  A  +  (-  B). 

By  this  means  subtraction  is  reduced  to  addition  and  needs 


12 


VECTOR  ANALYSIS 


no  special  consideration.    There  is  however  an  interesting  and 
important  way  of  representing  the  difference  of  two  vectors 

Complete 


geometrically. 


Let  A  =  OA,  B  =  OB  (Fig.  4). 

the  parallelogram  of  which  A  and  B 
are  the  sides.  Then  the  diagonal 
0  C  =  C  is  the  sum  A  +  B  of  the 
two  vectors.  Next  complete  the 
parallelogram  of  which  A  and  —  B 
—  OB'  are  the  sides.  Then  the  di- 


agonal OD  =  D  will  be  the  sum  of 
A  and  the  negative  of  B.  But  the 
segment  OD  is  parallel  and  equal 

to  EA.  Hence  BA  may  be  taken  as  the  difference  to  the  two 
vectors  A  and  B.  This  leads  to  the  following  rule  :  The  differ- 
ence of  two  vectors  which  are  drawn  from  the  same  origin  is 
the  vector  drawn  from  the  terminus  of  the  vector  to  be  sub- 
tracted to  the  terminus  of  the  vector  from  which  it  is  sub- 
tracted. Thus  the  two  diagonals  of  the  parallelogram,  which 
is  constructed  upon  A  and  B  as  sides,  give  the  sum  and  dif- 
ference of  A  and  B. 

12.]  In  the  foregoing  paragraphs  addition,  subtraction,  and 
scalar  multiplication  of  vectors  have  been  defined  and  inter- 
preted. To  make  the  development  of  vector  algebra  mathe- 
matically exact  and  systematic  it  would  now  become  necessary 
to  demonstrate  that  these  three  fundamental  operations  follow 
the  same  formal  laws  as  in  the  ordinary  scalar  algebra,  al- 
though from  the  standpoint  of  the  physical  and  geometrical 
interpretation  of  vectors  this  may  seem  superfluous.  These 

laws  are 

I  a      m  (n  A)      =  n  (m  A)  =  (m  ri)  A, 

I6  (A  +  B)  +  C  =  A+  (B-f  C), 

II  A  -I-  B  =  B  +  A, 

IIIa  (m,  +  n)  A  =  m  A  +  n  A, 

III6  m  (A  +  B)  =  m  A  +  m  B, 

III,  -  (A  +  B)  =  -  A  -  B. 


ADDITION  AND  SCALAR  MULTIPLICATION          13 

Ia  is  the  so-called  law  of  association  and  commutation  of 
the  scalar  factors  in  scalar  multiplication. 

I6  is  the  law  of  association  for  vectors  in  vector  addition.  It 
states  that  in  adding  vectors  parentheses  may  be  inserted  at 
any  points  without  altering  the  result. 

II  is  the  commutative  law  of  vector  addition. 

IIIa  is  the  distributive  law  for  scalars  in  scalar  multipli- 
cation. 

IIIj  is  the  distributive  law  for  vectors  in  scalar  multipli- 
cation. 

IIIC  is  the  distributive  law  for  the  negative  sign. 

The  proofs  of  these  laws  of  operation  depend  upon  those 
propositions  in  elementary  geometry  which  have  to  deal  with 
the  first  properties  of  the  parallelogram  and  similar  triangles. 
They  will  not  be  given  here;  but  it  is  suggested  that  the 
reader  work  them  out  for  the  sake  of  fixing  the  fundamental 
ideas  of  addition,  subtraction,  and  scalar  multiplication  more 
clearly  in  mind.  The  result  of  the  laws  may  be  summed  up 
in  the  statement : 

The  laws  which  govern  addition,  subtraction,  and  scalar 
multiplication  of  vectors  are  identical  with  those  governing  these 
operations  in  ordinary  scalar  algebra. 

It  is  precisely  this  identity  of  formal  laws  which  justifies 
the  extension  of  the  use  of  the  familiar  signs  =,  +,  and  — 
of  arithmetic  to  the  algebra  of  vectors  and  it  is  also  this 
which  ensures  the  correctness  of  results  obtained  by  operat- 
ing with  those  signs  in  the  usual  manner.  One  caution  only 
need  be  mentioned.  Scalars  and  vectors  are  entirely  different 
sorts  of  quantity.  For  this  reason  they  can  never  be  equated 
to  each  other  —  except  perhaps  in  the  trivial  case  where  each  is 
zero.  For  the  same  reason  they  are  not  to  be  added  together. 
So  long  as  this  is  borne  in  mind  no  difficulty  need  be  antici- 
pated from  dealing  with  vectors  much  as  if  they  were  scalars. 

Thus  from  equations  in  which  the  vectors  enter  linearly  with 


14  VECTOR  ANALYSIS 

scalar  coefficients  unknown  vectors  may  be  eliminated  or 
found  by  solution  in  the  same  way  and  with  the  same  limita- 
tions as  in  ordinary  algebra;  for  the  eliminations  and  solu- 
tions depend  solely  on  the  scalar  coefficients  of  the  equations 
and  not  at  all  on  what  the  variables  represent.  If  for 

instance 

aA  +  &B  +  cC  +  dD  =  0, 

then  A,  B,  C,  or  D  may  be  expressed  in  terms  of  the  other 

three 

as  D  =  -  -  (a  A  +  b  B  +  c  C). 

(M 

And  two  vector  equations  such  as 

3 A+4B=E 

and  2  A  +  3  B  =  F 

yield  by  the  usual  processes  the  solutions 

A=3E-4F 
and  B  =  3  F  -  2  E. 

Components  of  Vectors 

13.]  Definition :  Vectors  are  said  to  be  collinear  when 
they  are  parallel  to  the  same  line;  coplanar,  when  parallel 
to  the  same  plane.  Two  or  more  vectors  to  which  no  line 
can  be  drawn  parallel  are  said  to  be  non-collinear.  Three  or 
more  vectors  to  which  no  plane  can  be  drawn  parallel  are 
said  to  be  non-coplanar.  Obviously  any  two  vectors  are 
coplanar. 

Any  vector  b  collinear  with  a  may  be  expressed  as  the 
product  of  a  and  a  positive  or  negative  scalar  which  is  the 
ratio  of  the  magnitude  of  b  to  that  of  a.  The  sign  is  positive 
when  b  and  a  have  the  same  direction  ;  negative,  when  they 
have  opposite  directions.  If  then  OA  =  a,  the  vector  r  drawn 


ADDITION  AND  SCALAR  MULTIPLICATION          15 

from  the  origin  0  to  any  point  of  the  line  OA  produced  in 

either  direction  is 

r  =  x  a.  (1) 

If  x  be  a  variable  scalar  parameter  this  equation  may  there- 
fore be  regarded  as  the  (vector)  equation  of  all  points  in  the 
line  OA.     Let  now  B  be  any  point  not 
upon  the  line  OAor  that  line  produced 
in  either  direction  (Fig.  5). 

Let  0  B  =  b.     The  vector  b  is  surely 


o 


not  of  the  form  x  a.     Draw  through  B  FlG  5 

a  line  parallel  to  OA  and  let  E  be  any 

point  upon  it.      The  vector  BE  is  collinear  with  a  and  is 

consequently  expressible  as  x  a.      Hence  the  vector  drawn 

from  0  to  E  is 

0~R=  CTB  +  B~R 

or  r  =  b  +  «a.  (2) 

This  equation  may  be  regarded  as  the  (vector)  equation  of 
all  the  points  in  the  line  which  is  parallel  to  a  and  of  which 
B  is  one  point. 

14.]     Any  vector  r  coplanar  with  two  non-collinear  vectors 
a  and  b  may  be  resolved  into  two  components  parallel  to  a 
and  b  respectively.     This  resolution  may 
be  accomplished  by  constructing  the  par- 
allelogram (Fig.  6)  of  which  the  sides  are 
parallel  to  a  and  b  and  of  which  the  di- 
agonal is  r.     Of  these  components  one  is 
x  a ;  the  other,  y  b.     x  and  y  are  respec- 
tively the   scalar  ratios  (taken  with  the 
proper  sign)  of  the  lengths  of  these  components  to  the  lengths 

of  a  and  b.     Hence 

r  =  x  a  +  y  b  (2)' 

is  a  typical  form  for  any  vector  coplanar  with  a  and  b.  If 
several  vectors  rv  r2,  r3  •  •  •  may  be  expressed  in  this  form  as 


16 


VECTOR  ANALYSIS 


their  sum  r  is  then 


rx  =  xl  a  +  yl  b, 

r2  =  z2  a  +  2/2  b, 
r3  =  ^3  a  +  2/3  k 


+  (2/i  +  y2  +  2/3  +  ---)  b- 

This  is  the  well-known  theorem  that  the  components  of  a 
sum  of  vectors  are  the  sums  of  the  components  of  those 
vectors.  If  the  vector  r  is  zero  each  of  its  components  must 
be  zero.  Consequently  the  one  vector  equation  r  =  0  is 
equivalent  to  the  two  scalar  equations 


2/i  +  2/2  +  2/3  +  •  •  •  = 


(3) 


15.]     Any  vector  r  in  space  may  be  resolved  into  three 
components  parallel  to  any  three  given  non-coplanar  vectors. 

Let  the  vectors  be  a,  b, 
and  c.  The  resolution 
may  then  be  accom- 
plished by  constructing 
the  parallelepiped  (Fig. 
7)  of  which  the  edges 
are  parallel  to  a,  b,  and 
c  and  of  which  the  di- 
agonal is  r.  This  par- 
allelopiped  may  be 
drawn  easily  by  passing 
three  planes  parallel  re- 

spectively to  a  and  b,  b  and  c,  c  and  a  through  the  origin  0 
of  the  vector  r  ;  and  a  similar  set  of  three  planes  through  its 
terminus  R.  These  six  planes  will  then  be  parallel  in  pairs 


pIG  7 


ADDITION  AND  SCALAR  MULTIPLICATION          17 

and  hence  form  a  parallelepiped.  That  the  intersections  of 
the  planes  are  lines  which  are  parallel  to  a,  or  b,  or  c  is 
obvious.  The  three  components  of  r  are  x  a,  y  b,  and  z  c ; 
where  #,  y,  and  z  are  respectively  the  scalar  ratios  (taken  with 
the  proper  sign)  of  the  lengths  of  these  components  to  the 
length  of  a,  b,  and  c.  Hence 

r  =  #a  +  2/b  +  zc  (4) 

is  a  typical  form  for  any  vector  whatsoever  in  space.  Several 
vectors  iv  r2,  r3  .  .  .  may  be  expressed  in  this  form  as 

^  =  xl  a  +  yl  b  +  zl  c, 
r2  =  aja  a  +  y^  b  +  z2  c, 
r3  =  #3  a  +  yB  b  +  *3  c, 


Their  sum  r  is  then 


+  (.l/i  +  2/2  +  2/3  +  '  •  •)  b 

+  (2J+Z2  +  23+  •••)<>• 

If  the  vector  r  is  zero  each  of  its  three  components  is  zero. 
Consequently  the  one  vector  equation  r  =  0  is  equivalent  to 
the  three  scalar  equations 


yi  +  2/2  +  2/3  +  '  '  '  =  0         r  -  0.  (5) 

*1  +  *2  +  23   +    '  '  '  =  °  ' 

Should  the  vectors  all  be  coplanar  with  a  and  b,  all  the  com- 
ponents parallel  to  c  vanish.  In  this  case  therefore  the  above 
equations  reduce  to  those  given  before. 

16.]  If  two  equal  vectors  are  expressed  in  terms  of  the 
same  three  non-coplanar  vectors,  the  corresponding  scalar  co- 
efficients are  equal. 

2 


18  VECTOR  ANALYSIS 

Let  r  =  r', 

r'  =  x'  a  +  y'  b  +  z'  c, 

Then  x  =  x',     y  =  y',    z  =  z9. 

For      r  -  r'  =  0  =  (x  -  x')  a  +  (y  -  y'}  b  +  (z  -  z')  c. 
Hence  x  —  x'  =  0,     y  —  y'  =  0,     z  —  z'  =  0. 

But  this  would  not  be  true  if  a,  b,  and  c  were  coplanar.  In 
that  case  one  of  the  three  vectors  could  be  expressed  in  terms 

of  the  other  two  as 

c  =  m  a  +  n  b. 

Then    r  =rca  +  2/b  +  2c  =  (ic  +  m2)a+(2/  +  7i2)b, 
r'  =  x'  a  +  yr  b  +  z'  c  =  (x'  +  m  z')  a  +  (y'  +  n  z')  b, 
r  —  r'  =  [(x  +  m  z  )  —  (#'  +  m  2')]  a, 

+  [(y  +  nz)-(y'  +  nz')']  b  =  0. 

Hence  the  individual  components  of  r  —  r'  in  the  directions 

a  and  b  (supposed  different)  are  zero. 

r-i  o  i"i /*»  £i  O"  L\ _.   vfl  v  ~—  /y*     J^   />ri  *y' 

I    1    i     J  H     V  •*/     T^      //t    *•     tX/         ^^      //(>    « 

^  +  n  z  =  y'  +  n  z'. 

But  this  by  no  means  necessitates  a?,  y,  z  to  be  equal  respec- 
tively to  #',  y',  z'.  In  a  similar  manner  if  a  and  b  were  col- 
linear  it  is  impossible  to  infer  that  their  coefficients  vanish 
individually.  The  theorem  may  perhaps  be  stated  as  follows : 
In  case  two  equal  vectors  are  expressed  in  terms  of  one  vector, 
or  two  non-collinear  vectors,  or  three  non-coplanar  vectors,  the 
corresponding  scalar  coefficients  are  equal.  But  this  is  not  ne- 
cessarily true  if  the  two  vectors  be  collinear  ;  or  the  three  vectors, 
coplanar.  This  principle  will  be  used  in  the  applications 
(Arts.  18  et  seq.). 

The  Three  Unit  Vectors  i,  j,  k. 

17.]  In  the  foregoing  paragraphs  the  method  of  express- 
ing vectors  in  terms  of  three  given  non-coplanar  ones  has  been 
explained.  The  simplest  set  of  three  such  vectors  is  the  rect- 


ADDITION  AND  SCALAR  MULTIPLICATION 


19 


angular  system  familiar  in  Solid  Cartesian  Geometry.  This 
rectangular  system  may  however  be  either  of  two  very  distinct 
types.  In  one  case  (Fig.  8,  first  part)  the  £-axis  l  lies  upon 
that  side  of  the  X  Y-  plane  on  which  rotation  through  a  right 
angle  from  the  X-axis  to  the  F-axis  appears  counterclockwise 
or  positive  according  to  the  convention  adopted  in  Trigonome- 
try. This  relation  may  be  stated  in  another  form.  If  the  X- 
axis  be  directed  to  the  right  and  the  F-axis  vertically,  the 
^-axis  will  be  .directed  toward  the  observer.  Or  if  the  X- 
axis  point  toward  the  observer  and  the  F-axis  to  the  right, 
the  Z'-axis  will  point  upward.  Still  another  method  of  state- 

Z  Z 


Right-handed 


FIG.  8. 


Left-handed 


ment  is  common  in  mathematical  physics  and  engineering.  If 
a  right-handed  screw  be  turned  from  the  X-axis  to  the  F- 
axis  it  will  advance  along  the  (positive)  Z-axis.  Such  a  sys- 
tem of  axes  is  called  right-handed,  positive,  or  counterclock- 
wise.2 It  is  easy  to  see  that  the  F-axis  lies  upon  that  side  of 
the  Z  JT-plane  on  which  rotation  from  the  .Z-axis  to  the  X- 
axis  is  counterclockwise ;  and  the  X-axis,  upon  that  side  of 

1  By  the  X-,  Y-,  or  Z-axis  the  positive  half  of  that  axis  is  meant.     The  X  Y- 
plane  means  the  plane  which  contains  the  X-  and  Y-axis,  f.  e.,  the  plane  z  =  0. 

2  A  convenient  right-handed  system  and  one  which  is  always  available  consists 
of  the  thumb,  first  finger,  and  second  finger  of  the  right  hand.    If  the  thumb  and 
first  finger  be  stretched  out  from  the  palm  perpendicular  to  each  other,  and  if  the 
second  finger  be  bent  over  toward  the  palm  at  right  angles  to  first  finger,  a  right- 
handed  system  is  formed  by  the  fingers  taken  in  the  order  thumb,  first  finger, 
second  finger. 


20  VECTOR   ANALYSIS 

the  YZ-plaue  on  which  rotation  from  the  F-axis  to  the  Z- 
axis  is  counterclockwise.  Thus  it  appears  that  the  relation 
between  the  three  axes  is  perfectly  symmetrical  so  long  as  the 
same  cyclic  order  X  YZX  Y  is  observed.  If  a  right-handed 
screw  is  turned  from  one  axis  toward  the  next  it  advances 
along  the  third. 

In  the  other  case  (Fig.  8,  second  part)  the  Z'-axis  lies  upon 
that  side  of  the  X  F-plane  on  which  rotation  through  a  right 
angle  from  the  JT-axis  to  the  F-axis  appears  clockwise,  or  neg- 
ative. The  F-axis  then  lies  upon  that  side  of  the  ^"J^-plane 
on  which  rotation  from  the  ^"-axis  to  the  JT-axis  appears 
clockwise  and  a  similar  statement  may  be  made  concerning 
the  X-axis  in  its  relation  to  the  F^-plane.  In  this  case,  too, 
the  relation  between  the  three  axes  is  symmetrical  so  long 
as  the  same  cyclic  order  X  YZX  Y  is  preserved  but  it  is  just 
the  opposite  of  that  in  the  former  case.  If  a  left-handed  screw 
is  turned  from  one  axis  toward  the  next  it  advances  along 
the  third.  Hence  this  system  is  called  left-handed,  negative, 
or  clockwise.1 

The  two  systems  are  not  superposable.  They  are  sym- 
metric. One  is  the  image  of  the  other  as  seen  in  a 
mirror.  If  the  X-  and  F-axes  of  the  two  different  systems  be 
superimposed,  the  ^"-axes  will  point  in  opposite  directions. 
Thus  one  system  may  be  obtained  from  the  other  by  reversing 
the  direction  of  one  of  the  axes.  A  little  thought  will  show 
that  if  two  of  the  axes  be  reversed  in  direction  the  system  will 
not  be  altered,  but  if  all  three  be  so  reversed  it  will  be. 

Which  of  the  two  systems  be  used,  matters  little.  But  in- 
asmuch as  the  formulae  of  geometry  and  mechanics  differ 
slightly  in  the  matter  of  sign,  it  is  advisable  to  settle  once  for 
all  which  shall  be  adopted.  In  this  book  the  right-handed  or 
counterclockwise  system  will  be  invariably  employed. 

1  A  left-handed  system  may  be  formed  by  the  left  hand  just  as  a  right-handed 
one  was  formed  by  the  right. 


ADDITION  AND  SCALAR  MULTIPLICATION         21 

Definition :  The  three  letters  i,  j,  k  will  be  reserved  to  de- 
note three  vectors  of  unit  length  drawn  respectively  in  the 
directions  of  the  X-,  F-,  and  Z-  axes  of  a  right-handed  rectan- 
gular system. 

In  terms  of  these  vectors,  any  vector  may  be  expressed  as 

r  =  x'\  +  y\  +  ak.  (6) 

The  coefficients  x,  y,  z  are  the  ordinary  Cartesian  coordinates 
of  the  terminus  of  r  if  its  origin  be  situated  at  the  origin  of 
coordinates.  The  components  of  r  parallel  to  the  JT-,  F-,  and 
.Z-axes  are  respectively 

x  i,     y  j,    z  k. 

The  rotations  about  i  from  j  to  k,  about  j  from  k  to  i,  and 
about  k  from  i  to  j  are  all  positive. 

By  means  of  these  vectors  i,  j,  k  such  a  correspondence  is 
established  between  vector  analysis  and  the  analysis  in  Car- 
tesian coordinates  that  it  becomes  possible  to  pass  at  will 
from  either  one  to  the  other.  There  is  nothing  contradic- 
tory between  them.  On  the  contrary  it  is  often  desirable 
or  even  necessary  to  translate  the  formulae  obtained  by 
vector  methods  into  Cartesian  coordinates  for  the  sake  of 
comparing  them  with  results  already  known  and  it  is 
still  more  frequently  convenient  to  pass  from  Cartesian 
analysis  to  vectors  both  on  account  of  the  brevity  thereby 
obtained  and  because  the  vector  expressions  show  forth  the 
intrinsic  meaning  of  the  formulae. 

Applications 

*18.]  Problems  in  plane  geometry  may  frequently  be  solved 
easily  by  vector  methods.  Any  two  non-collinear  vectors  in 
the  plane  may  be  taken  as  the  fundamental  ones  in  terms  of 
which  all  others  in  that  plane  may  be  expressed.  The  origin 
may  also  be  selected  at  pleasure.  Often  it  is  possible  to 


22  VECTOR  ANALYSIS 

make  such  an  advantageous  choice  of  the  origin  and  funda- 
mental vectors  that  the  analytic  work  of  solution  is  materially 
simplified.  The  adaptability  of  the  vector  method  is  about 
the  same  as  that  of  oblique  Cartesian  coordinates  with  differ- 
ent scales  upon  the  two  axes. 

(Example?^}  The  line  which  joins  one  vertex  of  a  parallelo- 
gram to  the  middle  point  of  an  opposite  side  trisects  the  diag- 

O  -t  ii  O 

onal  (Fig.  9). 

Let  A  BCD  be  the  parallelogram,  BE  the  line  joining  the 

vertex  B  to  the  middle  point  E  of  the  side 

p —   ^^Jc    AD,  R  the  point  in  which  this  line  cuts  the 

Z^^-^/       diagonal  A  C.     To  show  A  R  is  one  third  of 

F          '         AC.     Choose  A  as  origin,  A  B  and  A  D  as  the 

two   fundamental  vectors  S   and  T.      Then 

A  C  is  the  sum  of  S  and  T.     Let  A~R  =  R.     To  show 

E  -  1  (S  +  T). 
B  =  A~R  =  AE  +  ER  =|  T  +  x  (S  -  |  T), 

where  x  is  the  ratio  of  ER  to  EB  —  an  unknown  scalar. 
And  E  =  y  (S  +  T), 


where  y  is  the  scalar  ratio  of  A  jl(  to  A  G  to  be  shown  equal 
toj. 

Hence  \  T  +  x  (S  -  i  T)  =  y  (S  +  T) 


or  »Sl 


Hence,  equating  corresponding  coefficients  (Art.  16), 


ADDITION  AND  SCALAR  MULTIPLICATION         23 
From  which  y  =  -. 

Inasmuch  as  x  is  also  g-  the  line  EB  must  be  trisected  as 
well  as  the  diagonal  A  C. 

Example  %j)  If  through  any  point  within  a  triangle  lines 
be  drawn  parallel  to  the  sides  the  sum  of  the  ratios  of  these 
lines  to  their  corresponding  sides  is  2. 

Let  A  B  C  be  the  triangle,  R  the  point  within  it.  Choose 
A  as  origin,  A  B  and  A  C  as  the  two  fundamental  vectors  S 
and  T.  Let 

2TB  =  R  =  mS+ wT.  (a) 

m  S  is  the  fraction  of  A  B  which  is  cut  off  by  the  line  through 
R  parallel  to  A  C.  The  remainder  of  A  B  must  be  the  frac- 
tion 1  —  m  S.  Consequently  by  similar  triangles  the  ratio  of 
the  line  parallel  to  A  G  to  the  line  A  C  itself  is  (1  —  m). 
Similarly  the  ratio  of  the  line  parallel  to  A  B  to  the  line  A  B 
itself  is  (1  —  n ).  Next  express  R  in  terms  of  S  and  T  —  S  the 
third  side  of  the  triangle.  Evidently  from  (a) 

E  =  (m  +  n)  S  +  n  (T  -  S). 

Hence  (m  +  n)  S  is  the  fraction  of  A  B  which  is  cut  off  by  the 
line  through  R  parallel  to  B  C.  Consequently  by  similar  tri- 
angles the  ratio  of  this  line  to  B  C  itself  is  (m  -f  n).  Adding 
the  three  ratios 

(1  —  w)  +  (1  —  TO)  +  (m  +  n)  =  2, 

and  the  theorem  is  proved. 

Example  3 :  If  from  any  point  within  a  parallelogram  lines 
be  drawn  parallel  to  the  sides,  the  diagonals  of  the  parallelo- 
grams thus  formed  intersect  upon  the  diagonal  of  the  given 
parallelogram. 

Let  A  B  CD  be  a  parallelogram,  R  a  point  within  it,  KM 
and  Z^Vtwo  lines  through  R  parallel  respectively  to  AB  and 

? 

N 


24  VECTOR  ANALYSIS 

AD,  the  points  K,  Z,  M>  N  lying  upon  the  sides  DA,AB, 
B  C,  CD  respectively.  To  show  that  the  diagonals  KN  and 
LM  of  the  two  parallelograms  KEND  and  LBMR  meet 
on  A  C.  \  Choose  A  as  origin,  A  B  and  A  D  as  the  two  funda- 
mental vectors  S  and  T.  Let 


and  let  P  be  the  point  of  intersection  of  KN  with  LM. 


Then  KN=KR  +  RN  =  m  S  +  (1  -  n)  T, 


Zlf  =  (1  —  m)  S  +  ?i  T, 


Hence  P  =  w  T  +  x  [m  S  +  (1  -  n)  T], 

and  P  =  mS  +  y  [(l-m)S  +  wT], 

Equating  coefficients, 

*« 


n 
By  solution,  x  = 


y  = 


m  +  n  —  1 

m 
m  +  7i  —  1 


Substituting  either  of  these  solutions  in  the  expression  for  P, 
the  result  is 


m  +  n  —  1 

which  shows  that  P  is  collinear  with  A  G. 

*  19.]  Problems  in  three  dimensional  geometry  may  be 
solved  in  essentially  the  same  manner  as  those  in  two  dimen- 
sions. In  this  case  there  are  three  fundamental  vectors  in 
terms  of  which  all  others  can  be  expressed.  The  method  of 
solution  is  analogous  to  that  in  the  simpler  case.  Two 


ADDITION  AND  SCALAR  MULTIPLICATION          25 

expressions  for  the  same  vector  are  usually  found.  The  co- 
efficients of  the  corresponding  terms  are  equated.  In  this  way 
the  equations  between  three  unknown  scalars  are  obtained 
from  which  those  scalars  may  be  determined  by  solution  and 
then  substituted  in  either  of  the  expressions  for  the  required 
vector.  The  vector  method  has  the  same  degree  of  adapta- 
bility as  the  Cartesian  method  in  which  oblique  axes  with 
different  scales  are  employed.  The  following  examples  like 
those  in  the  foregoing  section  are  worked  out  not  so  much  for 
their  intrinsic  value  as  for  gaining  a  familiarity  with  vectors. 

Example  1  :  Let  A  B  CD  be  a  tetrahedron  and  P  any 
point  within  it.  Join  the  vertices  to  P  and  produce  the  lines 
until  they  intersect  the  opposite  faces  in  A',  B\  C',  D'.  To 

show 

PA'       PB'       PC'       PD' 


A~A'       £~B'       ~GG'  n"  LTD' 


_ 

~      ' 


Choose  A  as  origin,  and  the  edges  A  B,  AC,  AD  as  the 
three  fundamental  vectors  B,  C,  D.     Let  the  vector  A  P  be 


P  =  AP=l"B  +  mC  +  ?i  D, 


A'  =  A  A'  =  kl  P  =  kl  (I  B  +  m  C  +  n  D). 


Also  A'  =  AA'  =  AB+  BA'. 


The  vector  BA'  is  coplanar  with  BC  =  C  —  B  and  BD  = 
D  —  B.     Hence  it  may  be  expressed  in  terms  of  them, 

A'  =  B  +  xl  (C  -  B)  +  yl  (D  -  B). 
Equating  coefficients        Jc1  m  =  xv 


Hence  kl  = 


I  +  m  +  n 


T  -        •*-  1  ~4  SI  N. 

and  — —  =  ~ —  =  1  —  (7  +  m  +  n). 

n  -i 


26  VECTOR  ANALYSIS 


In  like  manner  A  B'  =  #2  C  +  yz  D 


and  ~AB'  =  ~AB  '+  &£'=  B  +  kz  (P  -  B). 

Hence     «;2C  +  y2D  =  B  +  &2(/B  +  mC  +  wD- 
and  0  =  1  -f  kz  (I  -  1), 

xz=Jczm, 

7/2  =  &2  n. 
1 


Hence  &„  = 

and 


l-l 
PB'      &„-: 


"2 

In  the  same  way  it  may  be  shown  that 
PC'  ,PZ>' 


. 

Adding  the  four  ratios  the  result  is 

1  —  (l  +  m  +  n)  +  l  +  m+n  =  l. 

Example  2 :  To  find  a  line  which  passes  through  a  given 
point  and  cuts  two  given  lines  in  space. 

Let  the  two  lines  be  fixed  respectively  by  two  points  A 
and  B,  0  and  D  on  each.  Let  0  be  the  given  point.  Choose 
it  as  origin  and  let 


V=OD. 
Any  point  P  of  A  B  may  be  expressed  as 

P  =  OP=  OA  +  x  TB  =  A  +  x  (B  -  A). 
Any  point  Q  of  CD  may  likewise  be  written 


If  the  points  P  and  Q  lie  in  the  same  line  through  0,  P  and 
are.collinear.     That  is 


ADDITION  AND  SCALAR  MULTIPLICATION         27 

Before  it  is  possible  to  equate  coefficients  one  of    the  four 
vectors  must  be  expressed  in  terms  of  the  other  three. 

Let  J)  =  lA.  +  m'B  +  nC. 

Then  P  =  A  +  x  (B  -  A) 

=  z  [C  +  y  (IA.+  ra  B  +  nC  —  0)]. 
Hence  1  —  x  =  z  y  I, 

x  =  zym, 

0  =  z  [1  +  y  (n  -  1)J. 
m 


Hence  x  = 


I  +  m 
1 


I  +  m 

Substituting  in  P  and  Q 

_  I  A+  m  B 

I  +  m 


Either  of  these  may  be  taken  as  defining  a  line  drawn  from  0 
and  cutting  A  B  and 


Vector  Relations  independent  of  the  Origin 

20.]     Example  1:  To  divide  a  line  AB  in  a  given  ratio 
m  :  n  (Fig.  10). 

Choose   any  arbitrary  point  0  as  A^*^^  P 

origin.       Let  OA  =  L  and  OB  =  '&.  &/    Px^^Jv^ 

To  find  the  vector  P  =~OP  of  which  f^^^ B 

the  terminus  P  divides  A  B  in  the  °         FlQ  10 
ratio  m :  n. 

P=  OP=  Oil  +  -^—  ~AB  =  &+  —^—  (B-A). 

m  +  n  m  +  n 

n  A  +  m  B  /7, 

That  is,  P  = (<) 

m  +  n 


28  VECTOR  ANALYSIS 

The  components  of  P  parallel  to  A  and  B  are  in  inverse  ratio 
to  the  segments  A  P  and  PB  into  which  the  line  A  B  is 
divided  by  the  point  P.  If  it  should  so  happen  that  P  divided 
the  line  AB  externally,  the  ratio  AP  /  PB  would  be  nega- 
tive, and  the  signs  of  m,  and  n  would  be  opposite,  but  the 
formula  would  hold  without  change  if  this  difference  of  sign 
in  m  and  n  be  taken  into  account. 

Example  2  :  To  find  the  point  of  intersection  of  the  medians 
of  a  triangle. 

Choose  the  origin  0  at  random.  Let  A  BC  be  the  given 
triangle.  Let  OA  =  A,  OB  '=  B,  and  ~OC  =  C.  Let  A\  B\  0' 
be  respectively  the  middle  points  of  the  sides  opposite  the 
vertices  A,  B,  G.  Let  M  be  the  point  of  intersection  of  the 
medians  and  M  =  OM  the  vector  drawn  to  it.  Then 


and 


Assuming  that  0  has  been  chosen  outside  of  the  plane  of  the 
triangle  so  that  A,  B,  C  are  non-coplanar,  corresponding  coeffi- 
cients may  be  equated. 


2  x  =  2 


2 

Hence  x  =  y  = 


Hence 


ADDITION  AND  SCALAR  MULTIPLICATION          29 

The  vector  drawn  to  the  median  point  of  a  triangle  is  equal 
to  one  third  of  the  sum  of  the  vectors  drawn  to  trie  vertices. 

In  the  problems  of  which  the  solution  has  just  been  given 
the  origin  could  be  chosen  arbitrarily  and  the  result  is  in- 
dependent of  that  choice.  Hence  it  is  even  possible  to  disre- 
gard the  origin  entirely  and  replace  the  vectors  A,  B,  C,  etc., 
by  their  termini  A,B,C,  etc.  Thus  the  points  themselves 
become  the  subjects  of  analysis  and  the  formulae  read 

n  A  +  m  B 
m  +  n 

and  M=\(A  +  B+C). 

This  is  typical  of  a  whole  class  of  problems  soluble  by  vector 
methods.  In  fact  any  purely  geometric  relation  between  the 
different  parts  of  a  figure  must  necessarily  be  independent 
of  the  origin  assumed  for  the  analytic  demonstration.  In 
some  cases,  such  as  those  in  Arts.  18, 19,  the  position  of  the 
origin  may  be  specialized  with  regard  to  some  crucial  point 
of  the  figure  so  as  to  facilitate  the  computation ;  but  in  many 
other  cases  the  generality  obtained  by  leaving  the  origin  un- 
specialized  and  undetermined  leads  to  a  symmetry  which 
renders  the  results  just  as  easy  to  compute  and  more  easy 
to  remember. 

Theorem :  The  necessary  and  sufficient  condition  that  a 
vector  equation  represent  a  relation  independent  of  the  origin 
is  that  the  sum  of  the  scalar  coefficients  of  the  vectors  on 
one  side  of  the  sign  of  equality  is  equal  to  the  sum  of  the 
coefficients  of  the  vectors  upon  the  other  side.  Or  if  all  the 
terms  of  a  vector  equation  be  transposed  to  one  side  leaving 
zero  on  the  other,  the  sum  of  the  scalar  coefficients  must 
be  zero. 

Let  the  equation  written  in  the  latter  form  be 


30  VECTOR  ANALYSIS 

Change  the  origin  from  0  to  0  '  by  adding  a  constant  vector 
E  =  00'  to  each  of  the  vectors  A,  B,  C,  D  ----  The  equation 
then  becomes 

a  (A  +  E)  +  b  (B  +  B)  +  c  (C  +  E)  +  d  (D  +  R)  -f  •  •  .  =  0 
=  aA  +  5B  +  cC  +  dD  +  ...  +B  (a  +  j,  +  c  +  d  +  ...). 

If  this  is  to  be  independent  of  the  origin  the  coefficient  of  E 
must  vanish.  Hence 


That  this  condition  is  fulfilled  in  the  two  examples  cited 
is  obvious. 


m  +  n 

1  „_*_.+     » 


m  +  n       m  +  n 
If  M=|(A  +  B  +  C), 

i  _1    ,    I    ,1 
~  3  ^  3    7"  *." 

*  21.]  The  necessary  and  sufficient  condition  that  two 
vectors  satisfy  an  equation,  in  which  the  sum  of  the  scalar 
coefficients  is  zero,  is  that  the  vectors  be  equal  in  magnitude 
and  in  direction. 

First  let  a  A  +  I  B  =  0 

and  a  +  6  =  0. 

It  is  of  course  assumed  that  not  both  the  coefficients  a  and  I 
vanish.  If  they  did  the  equation  would  mean  nothing.  Sub- 
stitute the  value  of  a  obtained  from  the  second  equation  into 
the  first. 


=  B. 


ADDITION  AND  SCALAR  MULTIPLICATION         31 

Secondly  if  A  and  B  are  equal  in  magnitude  and  direction 

the  equation 

A-B  =  0 

subsists  between  them.     The  sum  of  the  coefficients  is  zero. 

The  necessary  and  sufficient  condition  that  three  vectors 
satisfy  an  equation,  in  which  the  sum  of  the  scalar  coefficients 
is  zero,  is  that  when  drawn  from  a  common  origin  they  termi- 
nate in  the  same  straight  line.1 

First  let  «A  +  &B  +  cC  =  0 

and  a  +  b  +  c  =  0. 

Not  all  the  coefficients  a,  b,  c,  vanish  or  the  equations 
would  be  meaningless.  Let  c  be  a  non-vanishing  coefficient. 
Substitute  the  value  of  a  obtained  from  the  second  equation 

into  the  first. 

-(&  +  c)A+&B  +  cC  =  0, 

or  c  (C-A)  =  &(A-B). 

Hence  the  vector  which  joins  the  extremities  of  C  and  A  is 
collinear  with  that  which  joins  the  extremities  of  A  and  B. 
Hence  those  three  points  A,  B,  C  lie  on  a  line.  Secondly 
suppose  three  vectors  A  =  OA,  B  =  OB,C=  00  drawn  from 
the  same  origin  0  terminate  in  a  straight  line.  Then  the 
vectors 

A~B  =  B  -  A  and  AC r=  C  -  A 

are  collinear.    Hence  the  equation 

(B  -  A)  =  x  (C  -  A) 

subsists.  The  sum  of  the  coefficients  on  the  two  sides  is 
the  same. 

The  necessary  and  sufficient  condition  that  an  equation, 
in  which  the  sum  of  the  scalar  coefficients  is  zero,  subsist 

1  Vectors  which  hare  a  common  origin  and  terminate  in  one  line  are  called  by 
Hamilton  "  termino-collinear." 


32  VECTOR  ANALYSIS 

between  four  vectors,  is  that  if  drawn  from  a  common  origin 
they  terminate  in  one  plane.1 

First  let  aA  +  6B  +  cC  +  dD  =  0 

and  a  +  b  +  c  +  d  =  Q. 

Let  d  be  a  non-vanishing  coefficient.     Substitute  the  value 
of  a  obtained  from  the  last  equation  into  the  first. 


or  d(D-A)  =  6  (A-B)  +  c  (A  -  C). 

The  line  A  D  is  coplanar  with  A  B  and  A  C.  Hence  all  four 
termini  A,  B,  (7,  D  of  A,  B,  C,  D  lie  in  one  plane.  Secondly 
suppose  that  the  termini  of  A,  B,  C,  D  do  lie  in  one  plane. 
Then  A~D  =  D  -  A,  ~AC  =  C  -  A,  and  AB  —  B  -  A  are  co- 
planar vectors.  One  of  them  may  be  expressed  in  terms  of 
the  other  two.  This  leads  to  the  equation 

I  (B  -  A)  +  m  (C  -  A)  +  n  (D  -  A)  =  0, 

where  Z,  TO,  and  n  are  certain  scalars.  The  sum  of  the  coeffi- 
cients in  this  equation  is  zero. 

Between  any  five  vectors  there  exists  one  equation  the  sum 
of  whose  coefficients  is  zero. 

Let  A,  B,  C,  D,  E   be   the    five   given    vectors.      Form   the 

differences 

E-A,     E-B,     E-C,     E-D. 

One  of  these  may  be  expressed  in  terms  of  the  other  three 
—  or  what  amounts  to  the  same  thing  there  must  exist  an 
equation  between  them. 

k  (E  -  A)  +  I  (E  -  B)  +  m  (E  -  C)  +  n  (E  -  D)  ==  0. 
The  sum  of  the  coefficients  of  this  equation  is  zero. 

1  Vectors  which  have  a  common  origin  and  terminate  in  one  plane  are  called 
by  Hamilton  "  termino-complanar.1' 


ADDITION  AND  SCALAR  MULTIPLICATION          33 

*22.]  The  results  of  the  foregoing  section  afford  simple 
solutions  of  many  problems  connected  solely  with  the  geo- 
metric properties  of  figures.  Special  theorems,  the  vector 
equations  of  lines  and  planes,  and  geometric  nets  in  two  and 
three  dimensions  are  taken  up  in  order. 

Example  1:    If  a  line  be  drawn  parallel  to  the  base  of  a 
triangle,  the  line  which  joins  the  opposite  vertex  to  the  inter- 
section of    the   diagonals   of    the 
trapezoid  thus  formed  bisects  the 
base  (Fig.  11). 

Let  ABC  be  the  triangle,  ED 
the  line  parallel  to  the  base  CB, 
G  the  point  of  intersection  of  the 
diagonals  EB  and  DC  of  the  tra- 
pezoid CBDE,  and  jP'the  intersec- 
tion of  A  G  with  CB.  To  show 

rIG.  11. 

that  F  bisects  CB.     Choose  the 

origin  at  random.  Let  the  vectors  drawn  from  it  to  the 
various  points  of  the  figure  be  denoted  by  the  corresponding 
Clarendons  as  usual.  Then  since  ED  is  by  hypothesis  paral- 
lel to  CB,  the  equation 

E  -  D  =  n  (C  -  B) 

holds  true.  The  sum  of  the  coefficients  is  evidently  zero  as 
it  should  be.  Rearrange  the  terms  so  that  the  equation 

takes  on  the  form 

E  —  W  C  =  D  —  w  B. 

The  vector  E  —  n  C  is  coplanar  with  E  and  C.  It  must  cut 
the  line  E  C.  The  equal  vector  D  —  n  B  is  coplanar  with  D 
and  B.  It  must  cut  the  line  DB.  Consequently  the  vector 
represented  by  either  side  of  this  equation  must  pass  through 
the  point  A.  Hence 

E  —  7iC  =  D  —  wB  =  a;A. 


34  VECTOR  ANALYSIS 

However  the  points  E,  C,  and  A  lie  upon  the  same  straight 
line.  Hence  the  equation  which  connects  the  vectors  E,  C, 
and  A  must  be  such  that  the  sum  of  its  coefficients  is  zero. 
This  determines  x  as  1  —  n. 

Hence  E  —  nC  =  D  -  n~B  =  (1  -  w)  A. 

By  another  rearrangement  and  similar  reasoning 

E  +  wB=D  +  wC=  (1  +  w)  G. 
Subtract  the  first  equation  from  the  second  : 

n  (B  +  C)  =  (1  +  n)  0  -  (1  -  n)  A. 

This  vector  cuts  BO  and  AG.  It  must  therefore  be  a 
multiple  of  F  and  such  a  multiple  that  the  sum  of  the  coeffi- 
cients of  the  equations  which  connect  B,  C,  and  F  or  G,  A, 
and  F  shall  be  zero. 

Hence    n  (B  +  C)  =  (1  +  n)  G  -  (1  -  »)  A  =  2  nJf. 

B  +  C 

Hence  F  =  > 

SB 

and  the  theorem  has  been  proved.  The  proof  has  covered 
considerable  space  because  each  detail  of  the  reasoning  has 
been  given.  In  reality,  however,  the  actual  analysis  has  con- 
sisted of  just  four  equations  obtained  simply  from  the  first. 

Example  2  :  To  determine  the  equations  of  the  line  and 
plane. 

Let  the  line  be  fixed  by  two  points  A  and  B  upon  it.  Let 
P  be  any  point  of  the  line.  Choose  an  arbitrary  origin. 
The  vectors  A,  B,  and  P  terminate  in  the  same  line.  Hence 


and  a  +  b  +  p  =  0. 

aA  +  6B 


Therefore  P  = 


a  +  6 


ADDITION  AND  SCALAR  MULTIPLICATION         35 

For  different  points  P  the  scalars  a  and  I  have  different 
values.  They  may  be  replaced  by  x  and  y,  which  are  used 
more  generally  to  represent  variables.  Then 

_  xA.  +  yE 
x  +  y 

Let  a  plane  be  determined  by  three  points  A,B,  and  C. 
Let  P  be  any  point  of  the  plane.  Choose  an  arbitrary  origin. 
The  vectors  A,  B,  C,  and  P  terminate  in  one  plane.  Hence 


aA+  SB  +  cC 
and  a  +  b  +  c+p  =  Q. 

aA.  +  JB  +  cC 


Therefore  P  = 


a  +  b  +  c 


As  a,  Z>,  c,  vary  for  different  points  of  the  plane,  it  is  more 
customary  to  write  in  their  stead  x,  y,  z. 


x  +  y  +  z 

Example  3 ;  The  line  which  joins  one  vertex  of  a  com- 
plete quadrilateral  to  the  intersection  of  two  diagonals 
divides  the  opposite  sides  har- 
monically (Fig.  12). 

Let  A,  B,  C,  D  be  four  vertices 
of  a  quadrilateral.  Let  A  B  meet 
CD  in  a  fifth  vertex  E,  and  A  D 
meet  BC  in  the  sixth  vertex  F. 
Let  the  two  diagonals  AC  and 
BD  intersect  in  G.  To  show 

that  FG  intersects  A  B  in  a  point  E'  and  CD  in  a  point  E1' 
such  that  the  lines  AB  and  CD  are  divided  internally  at 
E'  and  E"  in  the  same  ratio  as  they  are  divided  externally 
by  E.  That  is  to  show  that  the  cross  ratios 

(A  B  .  EE')  =  (CD  -EE")  =  - 1. 


36  VECTOR  ANALYSIS 

Choose  the  origin  at  random.     The  four  vectors  A,  B,  C,  D 
drawn  from  it  to  the  points  A,  B,  C\  D  terminate  in  one 

plane.    Hence 

aA  +  &B  +  cC  +  dD  =  0 

and  a-f-&  +  c  +  ^  =  0. 

Separate  the  equations  by  transposing  two  terms  : 

«A  +  cC  =  —  (Z>B 
a  +  c  =  —  (b  +  d). 

«A  +  cC      5B  + 


Divide :  0  = 


In  like  manner      F  = 
Form: 


a  +  c  b  +  d 

A  +  d"D       6B  +  cC 


a  +  d  b  +  c 

(a  +  c)0  —  (a  +  d)?  cG  — 


(a  +  c)  —  (a  +  d)          (a  +  c)  —  (a  +  d) 


or  =  cC-dJ>  =  ^ 

c  —  d  c  —  d 


Separate  the  equations  again  and  divide : 
«A  +  JB       cC  +  dD 


c  +  d 


=  E.  (&) 


Hence  E  divides  A  B  in  the  ratio  a  :  b  and  CD  in  the  ratio 
c  .•  <2.  But  equation  (a)  shows  that  J£"  divides  CD  in  the 
ratio  —  c :  d.  Hence  E  and  E"  divide  CD  internally  and 
externally  in  the  same  ratio.  Which  of  the  two  divisions  is 
internal  and  which  external  depends  upon  the  relative  signs 
of  c  and  d.  If  they  have  the  same  sign  the  internal  point 
of  division  is  E;  if  opposite  signs,  it  is  E".  In  a  similar  way 
E'  and  E  may  be  shown  to  divide  A  B  harmonically. 

Example  4  '•  To  discuss  geometric  nets. 

By  a  geometric  net  in  a  plane  is  meant  a  figure  composed 
of  points  and  straight  lines  obtained  in  the  following  manner. 
Start  with  a  certain  number  of  points  all  of  which  lie  in  one 


ADDITION  AND  SCALAR  MULTIPLICATION          37 

plane.  Draw  all  the  lines  joining  these  points  in  pairs. 
These  lines  will  intersect  each  other  in  a  number  of  points. 
Next  draw  all  the  lines  which  connect  these  points  in  pairs. 
This  second  set  of  lines  will  determine  a  still  greater  number 
of  points  which  may  in  turn  be  joined  in  pairs  and  so  on. 
The  construction  may  be  kept  up  indefinitely.  At  each  step 
the  number  of  points  and  lines  in  the  figure  increases. 
Probably  the  most  interesting  case  of  a  plane  geometric  net  is 
that  in  which  four  points  are  given  to  commence  with. 
Joining  these  there  are  six  lines  which  intersect  in  three 
points  different  from  the  given  four.  Three  new  lines  may 
now  be  drawn  in  the  figure.  These  cut  out  six  new  points. 
From  these  more  lines  may  be  obtained  and  so  on. 

To  treat  this  net  analytically  write  down  the  equations 

aA  +  6B  +  cC  +  dD  =  0  (c) 

and  a  +  b  +  c  +  d  =  Q 

which  subsist  between  the  four  vectors  drawn  from  an  unde- 
termined origin  to  the  four  given  points.  From  these  it  is 

possible  to  obtain 

«A  +  b"B       cC  =  dD 
E  = 


a  +  b  c  +  d 

F  =  o — +_c_  _ 


b  +  d 
"B  +  cG 


a  +  d  b  +  c 

by  splitting  the  equations  into  two  parts  and  dividing.  Next 
four  vectors  such  as  A,  D,  E,  F  may  be  chosen  and  the  equa- 
tion the  sum  of  whose  coefficients  is  zero  may  be  determined. 
This  would  be 

—  aA.  +  d'D  +  (a  +  b)'E+  (a  +  c)  F  =  0. 

By  treating  this  equation  as  (c)  was  treated  new  points  may 
be  obtained. 


38  VECTOR  ANALYSIS 

—  a  A  +  d  D       (a  +  6)  E  +  (a  +  c) ! 


1  = 


—  a  +  d  ~  2a  +  b  +  c 

-  a  A  +  (a  +  &)  E  _  d  D  +  (a  +  c)  F 
6  a  +  c  +  d 

dD  +  (a  +  6)E 


c  a  +  b  +  d 

Equations  between  other  sets  of  four  vectors  selected  from 
A, B,C, D, E, F, G  may  be  found;  and  from  these  more  points 
obtained.  The  process  of  finding  more  points  goes  forward 
indefinitely.  A  fuller  account  of  geometric  nets  may  be 
found  in  Hamilton's  "  Elements  of  Quaternions,"  Book  I. 

As  regards  geometric  nets  in  space  just  a  word  may  be 
said.  Five  points  are  given.  From  these  new  points  may  be 
obtained  by  finding  the  intersections  of  planes  passed  through 
sets  of  three  of  the  given  points  with  lines  connecting  the 
remaining  pairs.  The  construction  may  then  be  carried  for- 
ward with  the  points  thus  obtained.  The  analytic  treatment 
is  similar  to  that  in  the  case  of  plane  nets.  There  are 
five  vectors  drawn  from  an  undetermined  origin  to  the  given 
five  points.  Between  these  vectors  there  exists  an  equation 
the  sum  of  whose  coefficients  is  zero.  This  equation  may  be 
separated  into  parts  as  before  and  the  new  points  may  thus 
be  obtained. 

aA  +  b"B      cG  +  dD  +  «E 


then 


a  +  b  c  +  d  +  e 


a  +  b  b  +  d  +  c 


are  two  of  the  points  and  others  may  be  found  in  the  same 
way.    Nets  in  space  are  also  discussed  by  Hamilton,  loc.  cit. 


ADDITION  AND  SCALAR  MULTIPLICATION          39 

Centers  of  Gravity 

*  23.]  The  center  of  gravity  of  a  system  of  particles  may 
be  found  very  easily  by  vector  methods.  The  two  laws  of 
physics  which  will  be  assumed  are  the  following: 

1°.  The  center  of  gravity  of  two  masses  (considered  as 
situated  at  points)  lies  on  the  line  connecting  the  two  masses 
and  divides  it  into  two  segments  which  are  inversely  pro- 
portional to  the  masses  at  the  extremities. 

2°.  In  finding  the  center  of  gravity  of  two  systems  of 
masses  each  system  may  be  replaced  by  a  single  mass  equal 
in  magnitude  to  the  sum  of  the  masses  in  the  system  and 
situated  at  the  center  of  gravity  of  the  system. 

Given  two  masses  a  and  b  situated  at  two  points  A  and  B. 
Their  center  of  gravity  G  is  given  by 


where  the  vectors  are  referred  to  any  origin  whatsoever. 
This  follows  immediately  from  law  1  and  the  formula  (7) 
for  division  of  a  line  in  a  given  ratio. 

The  center  of  gravity  of  three  masses  a,  o,  c  situated  at  the 
three  points  A,  B,  C  may  be  found  by  means  of  law  2.  The 
masses  a  and  b  may  be  considered  as  equivalent  to  a  single 
mass  a  -f  b  situated  at  the  point 

a  A  +  &B 


a  +  b 


Then  G  =  (a  +  6)  +  c  C 

a  +  0 


a  +  b  +  c 
a  A  +  &B  +  cC 


Hence  G  = 

a  +  b  +  c 


40  VECTOR  ANALYSIS 

Evidently  the  center  of  gravity  of  any  number  of  masses 
a,  6,  c,  d,  .  .  .  situated  at  the  points  A,  JB,  C,  Z>,  ...  may 
be  found  in  a  similar  manner.  The  result  is 

_  aA  +  &B  +  cC  +  ^D  +  ••• 
a  +  l  +  c  +  d+  ... 

Theorem  1  :  The  lines  which  join  the  center  of  gravity  of  a 
triangle  to  the  vertices  divide  it  into  three  triangles  which 
are  proportional  to  the  masses  at  the  op- 
posite vertices  (Fig.  13).  Let  A,B,C 
be  the  vertices  of  a  triangle  weighted 
with  masses  a,  Z>,  c.  Let  G  be  the  cen- 
ter of  gravity.  Join  A,  B,  C  to  G  and 
produce  the  lines  until  they  intersect 
the  opposite  sides  hi  A',  2?',  C"  respectively.  To  show  that 
the  areas 

GBC:GCA:GAB:ABC=a,:l:c:a  +  l  +  c. 

The  last  proportion  between  ABC  and  a  +  I  +  c  comes 
from  compounding  the  first  three.  It  is,  however,  useful  in 
the  demonstration. 

ABC      A  A'       AG      GA'  _b  +  c 

=    ~~ 


Hence 


G  B  C  ~  G  A' 
ABC 


In  a  similar  manner 
and 


GBC  a 

BCA       a  +  J  4-  c 


GCA  ~  b 

CAB  _  a  +  I  +  c 
GAB=       ~c         ' 


Hence  the  proportion  is  proved. 

Theorem  2 :   The  lines  which  join  the  center  of  gravity  of 
a  tetrahedron  to  the  vertices  divide  the  tetrahedron  into  four 


ADDITION  AND  SCALAR  MULTIPLICATION          41 

0 

tetrahedra  which  are  proportional  to  the  masses  at  the  oppo- 
site vertices. 

Let  A,  B,  C,  D  be  the  vertices  of  the  tetrahedron  weighted 
respectively  with  weights  a,  b,  c,  d.  Let  G  be  the  center  of 
gravity.  Join  A,  B,  C,  D  to  G  and  produce  the  lines  until 
they  meet  the  opposite  faces  in  A',  £',  C\  D'.  To  show  that 
the  volumes 

BCDG:CDAG:DABG:ABCG:ABGD 

=  a  :  b  :  c  :  d  :  a  +  b  +  c  +  d. 

BCD  A       A  A'       A  G        G  A'       b  +  c  +  d 


BCDG  ~  ~G~A7       ~GAJ    "  ~GA' 
a  +  b  +  c  +  d 


In  like  manner 

and 

and 


a 

CD  A  G  _  a  +  b  +  c  +  d 
CDAB~  b       "' 

DABG _a  +  b  +  c  +  d 
DABC  =  c         ~ ' 

ABCG _ a+b+c+d 
A  BCD'          ~d~     "' 


which  proves  the  proportion. 

*  24.]  By  a  suitable  choice  of  the  three  masses,  a,  6,  c  lo- 
cated at  the  vertices  A,  B,  C,  the  center  of  gravity  G  may 
be  made  to  coincide  with  any  given  point  P  of  the  triangle. 
If  this  be  not  obvious  from  physical  considerations  it  cer- 
tainly becomes  so  in  the  light  of  the  foregoing  theorems. 
For  in  order  that  the  center  of  gravity  fall  at  P,  it  is  only 
necessary  to  choose  the  masses  a,  &,  c  proportional  to  the 
areas  of  the  triangles  PBC,  PC  A,  and  PAB  respectively. 
Thus  not  merely  one  set  of  masses  a,  &,  c  may  be  found,  but 
an  infinite  number  of  sets  which  differ  from  each  other  only 
by  a  common  factor  of  proportionality.  These  quantities 


42  VECTOR  ANALYSIS 

a,  5,  c  may  therefore  be  looked  upon  as  coordinates  of  the 
points  P  inside  of  the  triangle  ABC.  To  each  set  there 
corresponds  a  definite  point  P,  and  to  each  point  P  there 
corresponds  an  infinite  number  of  sets  of  quantities,  which 
however  do  not  differ  from  one  another  except  for  a  factor 
of  proportionality. 

To  obtain  the  points  P  of  the  plane  ABC  which  lie  outside 
of  the  triangle  ABC  one  may  resort  to  the  conception  of 
negative  weights  or  masses.  The  center  of  gravity  of  the 
masses  2  and  —  1  situated  at  the  points  A  and  B  respectively 
would  be  a  point  G  dividing  the  line  A  B  externally  in  the 
ratio  1  :  2.  That  is 


Any  point  of  the  line  A  B  produced  may  be  represented  by 
a  suitable  set  of  masses  a,  b  which  differ  in  sign.  Similarly 
any  point  P  of  the  plane  ABO  may  be  represented  by  a 
suitable  set  of  masses  a,  Z>,  c  of  which  one  will  differ  in  sign 
from  the  other  two  if  the  point  P  lies  outside  of  the  triangle 
ABC.  Inasmuch  as  only  the  ratios  of  a,  Z>,  and  c  are  im- 
portant two  of  the  quantities  may  always  be  taken  positive. 

The  idea  of  employing  the  masses  situated  at  the  vertices 
as  coordinates  of  the  center  of  gravity  is  due  to  Mobius  and 
was  published  by  him  in  his  book  entitled  "  Barycentrische 
Calcul"  in  1826.  This  may  be  fairly  regarded  as  the  starting 
point  of  modern  analytic  geometry. 

The  conception  of  negative  masses  which  have  no  existence 
in  nature  may  be  avoided  by  replacing  the  masses  at  the 
vertices  by  the  areas  of  the  triangles  GBC,  GCA,  and 
GAB  to  which  they  are  proportional.  The  coordinates  of 
a  point  P  would  then  be  three  numbers  proportional  to  the 
areas  of  the  three  triangles  of  which  P  is  the  common  vertex  ; 
and  the  sides  of  a  given  triangle  ABC,  the  bases.  The  sign 
of  these  areas  is  determined  by  the  following  definition. 


ADDITION  AND  SCALAR  MULTIPLICATION          43 

Definition:  The  area  ABC  of  a  triangle  is  said  to  be 
positive  when  the  vertices  A,  B,  C  follow  each  other  in  the 
positive  or  counterclockwise  direction  upon  the  circle  de- 
scribed through  them.  The  area  is  said  to  be  negative  when 
the  points  follow  in  the  negative  or  clockwise  direction. 

Cyclic  permutation  of  the  letters  therefore  does  not  alter 
the  sign  of  the  area. 

ABG  =  BCA  =  CAB. 

Interchange  of  two  letters  which  amounts  to  a  reversal  of 
the  cyclic  order  changes  the  sign. 


If  P  be  any  point  within  the  triangle  the  equation 
PA  B  +  PB  C  +  PGA  =  A  B  C 

must  hold.  The  same  will  also  hold  if  P  be  outside  of  the 
triangle  provided  the  signs  of  the  areas  be  taken  into  con- 
sideration. The  areas  or  three  quantities  proportional  to 
them  may  be  regarded  as  coordinates  of  the  point  P. 

The  extension  of  the  idea  of  "  larycentric  "  coordinates  to 
space  is  immediate.  The  four  points  A,  B,  C,  D  situated  at 
the  vertices  of  a  tetrahedron  are  weighted  with  mass  a,  &,  c,  d 
respectively.  The  center  of  gravity  Gr  is  represented  by 
these  quantities  or  four  others  proportional  to  them.  To 
obtain  points  outside  of  the  tetrahedron  negative  masses 
may  be  employed.  Or  in  the  light  of  theorem  2,  page  40, 
the  masses  may  be  replaced  by  the  four  tetrahedra  which 
are  proportional  to  them.  Then  the  idea  of  negative  vol- 
umes takes  the  place  of  that  of  negative  weights.  As  this 
idea  is  of  considerable  importance  later,  a  brief  treatment  of 
it  here  may  not  be  out  of  place. 

Definition  :  The  volume  A  B  CD  of  a  tetrahedron  is  said 
to  be  positive  when  the  triangle  ABC  appears  positive  to 


44  VECTOR  ANALYSIS 

the  eye  situated  at  the  point  D.  The  volume  is  negative 
if  the  area  of  the  triangle  appear  negative. 

To  make  the  discussion  of  the  signs  of  the  various 
tetrahedra  perfectly  clear  it  is  almost  necessary  to  have  a 
solid  model.  A  plane  drawing  is  scarcely  sufficient.  It  is 
difficult  to  see  from  it  which  triangles  appear  positive  and 
which  negative.  The  following  relations  will  be  seen  to 
hold  if  a  model  be  examined. 

The  interchange  of  two  letters  in  the  tetrahedron  A  B  CD 
changes  the  sign. 

ACBD=CBAD=BACD=DBCA 


The  sign  of  the  tetrahedron  for  any  given  one  of  the  pos- 
sible twenty-four  arrangements  of  the  letters  may  be  obtained 
by  reducing  that  arrangement  to  the  order  A  B  0  D  by 
means  of  a  number  of  successive  interchanges  of  two  letters. 
If  the  number  of  interchanges  is  even  the  sign  is  the  same 
as  that  of  AB  CD  ;  if  odd,  opposite.  Thus 

CADB  =  -CABD  =  +  A  CBD  =  -ABCD. 

If  P  is  any  point  inside  of  the  tetrahedron  ABCD  the 
equation 

ABCP-BCDP+  CDAP-DABP=ABCD 

holds  good.  It  still  is  true  if  P  be  without  the  tetrahedron 
provided  the  signs  of  the  volumes  be  taken  into  considera- 
tion. The  equation  may  be  put  into  a  form  more  symmetri- 
cal and  more  easily  remembered  by  transposing  all  the  terms 
to  one  number.  Then 


ABCD  +  BCDP  +  CDPA  +  DPAB+PABC=Q. 

The  proportion  in  theorem  2,  page  40,  does  not  hold  true 
if  the  signs  of  the  tetrahedra  be  regarded.     It  should  read 

BCDG:CDGA:DOAB:GABC:ABCD 

=  a  :  b  :  c  :  d  :  a  -f  b  +  c  +  d. 


ADDITION  AND  SCALAR  MULTIPLICATION         45 

If  the  point  G-  lies  inside  the  tetrahedron  a,  5,  c,  d  repre- 
sent quantities  proportional  to  the  masses  which  must  be 
located  at  the  vertices  A,B,C,D  respectively  if  G  is  to  be  the 
center  of  gravity.  If  G  lies  outside  of  the  tetrahedron  they  may 
still  be  regarded  as  masses  some  of  which  are  negative  —  or 
perhaps  better  merely  as  four  numbers  whose  ratios  determine 
the  position  of  the  point  G-.  In  this  manner  a  set  of  "Jary- 
centric  "  coordinates  is  established  for  space. 

The  vector  P  drawn  from  an  indeterminate  origin  to  any 
point  of  the  plane  A  B  G  is  (page  35) 

_  a?A  +  yB  +  aC 


+  y  +  « 
Comparing  this  with  the  expression 

_  «A  +  6B  +  cC 
a  +  b  +  c 

it  will  be  seen  that  the  quantities  05,  yt  z  are  in  reality  nothing 
more  nor  less  than  the  barycentric  coordinates  of  the  point  P 
with  respect  to  the  triangle  ABO.  In  like  manner  from 

equation 

_xA.  +  y'B  +  zG  +  w'D 

x  +  y  +  z  +  w 

which  expresses  any  vector  P  drawn  from  an  indeterminate 
origin  in  terms  of  four  given  vectors  A,  B,  C,  D  drawn  from 
the  same  origin,  it  may  be  seen  by  comparison  with 

aA.  +  l'B  +  c  C  +  d"D 


b  +  c  +  d 


that  the  four  quantities  #,  y,  z,  w  are  precisely  the  bary- 
centric coordinates  of  P,  the  terminus  of  P,  with  respect  to 
the  tetrahedron  A  B  CD.  Thus  the  vector  methods  in  which 
the  origin  is  undetermined  and  the  methods  of  the  "  Bary- 
centric Calculus  "  are  practically  co-extensive. 
It  was  mentioned  before  and  it  may  be  well  to  repeat  here 


46 


VECTOR  ANALYSIS 


that  the  origin  may  be  left  wholly  out  of  consideration  and 
the  vectors  replaced  by  their  termini.  The  vector  equations 
then  become  point  equations 

xA  +  y  B  +  zC 


and 


P  = 


x  +  y  +  z 
xA  +  yB  +  z C  + wD 


At  0 


x  +  y  +  z  +  w. 

This  step  brings  in  the  points  themselves  as  the  objects  of 
analysis  and  leads  still  nearer  to  the  "  Barycentrische  Calcul " 
of  Mb'bius  and  the  "  Ausdehnungslehre  "  of  Grassmann. 

The   Use  of  Vectors  to  denote  Areas 

25.]  Definition :  An  area  lying  in  one  plane  M N  and 
bounded  by  a  continuous  curve  PQR  which  nowhere  cuts 
itself  is  said  to  appear  positive  from  the  point  0  when  the 

letters  PQR  follow  each 
other  in  the  counterclockwise 
or  positive  order;  negative, 
when  they  follow  in  the 
negative  or  clockwise  order 
(Fig.  14). 

It  is  evident  that  an  area 

Cf  can  have  no  determined  sign 

FIG.  14.  per  se^  but  only  in  reference 

to  that  direction  in  which  its 

boundary  is  supposed  to  be  traced  and  to  some  point  0  out- 
side of  its  plane.  For  the  area  P  R  Q  is  negative  relative  to 
PQR;  and  an  area  viewed  from  0  is  negative  relative  to  the 
same  area  viewed  from  a  point  0'  upon  the  side  of  the  plane 
opposite  to  0.  A  circle  lying  in  the  JTF-plane  and  described 
in  the  positive  trigonometric  order  appears  positive  from  every 
point  on  that  side  of  the  plane  on  which  the  positive  Z-axis 
lies,  but  negative  from  all  points  on  the  side  upon  which 


ADDITION  AND  SCALAR  MULTIPLICATION         47 

the  negative  ^-axis  lies.  For  this  reason  the  point  of  view 
and  the  direction  of  description  of  the  boundary  must  be  kept 
clearly  in  mind. 

Another  method  of  stating  the  definition  is  as  follows :  If 
a  person  walking  upon  a  plane  traces  out  a  closed  curve,  the 
area  enclosed  is  said  to  be  positive  if  it  lies  upon  his  left- 
hand  side,  negative  if  upon  his  right.  It  is  clear  that  if  two 
persons  be  considered  to  trace  out  together  the  same  curve  by 
walking  upon  opposite  sides  of  the  plane  the  area  enclosed 
will  lie  upon  the  right  hand  of  one  and  the  left  hand  of  the 
other.  To  one  it  will  consequently  appear  positive ;  to  the 
other,  negative.  That  side  of  the  plane  upon  which  the  area 
seems  positive  is  called  the  positive  side ;  the  side  upon 
which  it  appears  negative,  the  negative  side.  This  idea  is 
familiar  to  students  of  electricity  and  magnetism.  If  an 
electric  current  flow  around  a  closed  plane  curve  the  lines  of 
magnetic  force  through  the  circuit  pass  from  the  negative  to 
the  positive  side  of  the  plane.  A  positive  magnetic  pole 
placed  upon  the  positive  side  of  the  plane  will  be  repelled  by 
the  circuit. 

A  plane  area  may  be  looked  upon  as  possessing  more  than 
positive  or  negative  magnitude.  It  may  be  considered  to 
possess  direction,  namely,  the  direction  of  the  normal  to  the 
positive  side  of  the  plane  in  which  it  lies.  Hence  a  plane 
area  is  a  vector  quantity.  The  following  theorems  concerning 
areas  when  looked  upon  as  vectors  are  important. 

Theorem  1 :  If  a  plane  area  be  denoted  by  a  vector  whose 
magnitude  is  the  numerical  value  of  that  area  and  whose 
direction  is  the  normal  upon  the  positive  side  of  the  plane, 
then  the  orthogonal  projection  of  that  area  upon  a  plane 
will  be  represented  by  the  component  of  that  vector  in  the 
direction  normal  to  the  plane  of  projection  (Fig.  15). 

Let  the  area  A  lie  in  the  plane  MN.  Let  it  be  projected 
orthogonally  upon  the  plane  M'  N'.  Let  MN  and  M' N'  inter- 


48 


VECTOR  ANALYSIS 


sect  in  the  line  I  and  let  the  diedral  angle  between  these 
two  planes  be  x.  Consider  first  a  rectangle  PQRS  in  M N 
whose  sides,  PQ,  RS  and  QR,  SP  are  respectively  parallel 
and  perpendicular  to  the  line  I.  This  will  project  into  a 
rectangle  P'Q'R'S'  in  M'N'.  The  sides  P' Q'  and  R' S' 
will  be  equal  to  PQ  and  RS;  but  the  sides  Q' R'  and  S'P' 
will  be  equal  to  QR  and  SP  multiplied  by  the  cosine  of  #, 
the  angle  between  the  planes.  Consequently  the  rectangle 

P'Q'R'S'  =  PQRS  coax. 


FIG.  15. 

Hence  rectangles,  of  which  the  sides  are  respectively 
parallel  and  perpendicular  to  I,  the  line  of  intersection  of  the 
two  planes,  project  into  rectangles  whose  sides  are  likewise 
respectively  parallel  and  perpendicular  to  I  and  whose  area  is 
equal  to  the  area  of  the  original  rectangles  multiplied  by  the 
cosine  of  the  angle  between  the  planes. 

From  this  it  follows  that  any  area  A  is  projected  into  an 
area  which  is  equal  to  the  given  area  multiplied  by  the  cosine 
of  the  angle  between  the  planes.  For  any  area  A  may  be  di- 
vided up  into  a  large  number  of  small  rectangles  by  drawing  a 
series  of  lines  in  M N  parallel  and  perpendicular  to  the  line  /. 


ADDITION  AND  SCALAR  MULTIPLICATION          49 

Each  of  these  rectangles  when  projected  is  multiplied  by  the 
cosine  of  the  angle  between  the  planes  and  hence  the  total 
area  is  also  multiplied  by  the  cosine  of  that  angle.  On  the 
other  hand  the  component  A'  of  the  vector  A,  which  repre- 
sents the  given  area,  in  the  direction  normal  to  the  plane 
M'N'  of  projection  is  equal  to  the  total  vector  A  multiplied 
by  the  cosine  of  the  angle  between  its  direction  which  is 
the  normal  to  the  plane  MN  and  the  normal  to  M'N'.  This 
angle  is  x ;  for  the  angle  between  the  normals  to  two  planes 
is  the  same  as  the  angle  between  the  planes.  The  relation 
between  the  magnitudes  of  A  and  A'  is  therefore 

A'  =  A  cos  ic, 

which  proves  the  theorem. 

26.]  Definition :  Two  plane  areas  regarded  as  vectors  are 
said  to  be  added  when  the  vectors  which  represent  them  are 
added. 

A  vector  area  is  consequently  the  sum  of  its  three  com- 
ponents obtainable  by  orthogonal  projection  upon  three 
mutually  perpendicular  planes.  Moreover  in  adding  two 
areas  each  may  be  resolved  into  its  three  components,  the 
corresponding  components  added  as  scalar  quantities,  and 
these  sums  compounded  as  vectors  into  the  resultant  area. 
A  generalization  of  this  statement  to  the  case  where  the  three 
planes  are  not  mutually  orthogonal  and  where  the  projection 
is  oblique  exists. 

A  surface  made  up  of  several  plane  areas  may  be  repre- 
sented by  the  vector  which  is  the  sum  of  all  the  vectors 
representing  those  areas.  In  case  the  surface  be  looked  upon 
as  forming  the  boundary  of  a  portion  of  the  boundary  of  a 
solid,  those  sides  of  the  bounding  planes  which  lie  outside  of 
the  body  are  conventionally  taken  to  be  positive.  The  vec- 
tors which  represent  the  faces  of  solids  are  always  directed 
out  from  the  solid,  not  into  it. 

4 


50  VECTOR  ANALYSIS 

Theorem  2 :  The  vector  which  represents  a  closed  polyhedral 
surface  is  zero. 

This  may  be  proved  by  means  of  certain  considerations  of 
hydrostatics.  Suppose  the  polyhedron  drawn  in  a  body  of 
fluid  assumed  to  be  free  from  all  external  forces,  gravity  in- 
cluded.1 The  fluid  is  in  equilibrium  under  its  own  internal 
pressures.  The  portion  of  the  fluid  bounded  by  the  closed 
surface  moves  neither  one  way  nor  the  other.  Upon  each  face 
of  the  surface  the  fluid  exerts  a  definite  force  proportional 
to  the  area  of  the  face  and  normal  to  it.  The  resultant  of  all 
these  forces  must  be  zero,  as  the  fluid  is  in  equilibrium.  Hence 
the  sum  of  all  the  vector  areas  in  the  closed  surface  is  zero. 

The  proof  may  be  given  in  a  purely  geometric  manner. 
Consider  the  orthogonal  projection  of  the  closed  surface  upon 
any  plane.  This  consists  of  a  double  area.  The  part  of  the 
surface  farthest  from  the  plane  projects  into  positive  area ; 
the  part  nearest  the  plane,  into  negative  area.  Thus  the 
surface  projects  into  a  certain  portion  of  the  plane  which  is 
covered  twice,  once  with  positive  area  and  once  with  negative. 
These  cancel  each  other.  Hence  the  total  projection  of  a 
closed  surface  upon  a  plane  (if  taken  with  regard  to  sign)  is 
zero.  But  by  theorem  1  the  projection  of  an  area  upon  a 
plane  is  equal  to  the  component  of  the  vector  representing 
that  area  in  the  direction  perpendicular  to  that  plane.  Hence 
the  vector  which  represents  a  closed  surface  has  no  component 
along  the  line  perpendicular  to  the  plane  of  projection.  This, 
however,  was  any  plane  whatsoever.  Hence  the  vector  is 
zero. 

The  theorem  has  been  proved  for  the  case  in  which  the 
closed  surface  consists  of  planes.  In  case  that  surface  be 

1  Such  a  state  of  affairs  is  realized  to  all  practical  purposes  in  the  case  of  a 
polyhedron  suspended  in  the  atmosphere  and  consequently  subjected  to  atmos- 
pheric pressure.  The  force  of  gravity  acts  but  is  counterbalanced  by  the  tension 
in  the  suspending  string. 


ADDITION  AND  SCALAR  MULTIPLICATION          51 

curved  it  may  be  regarded  as  the  limit  of  a  polyhedral  surface 
whose  number  of  faces  increases  without  limit.  Hence  the 
vector  which  represents  any  closed  surface  polyhedral  or 
curved  is  zero.  If  the  surface  be  not  closed  but  be  curved  it 
may  be  represented  by  a  vector  just  as  if  it  were  polyhedral. 
That  vector  is  the  limit 1  approached  by  the  vector  which 
represents  that  polyhedral  surface  of  which  the  curved  surface 
is  the  limit  when  the  number  of  faces  becomes  indefinitely 
great. 

SUMMARY  OF  CHAPTER  I 

A  vector  is  a  quantity  considered  as  possessing  magnitude 
and  direction.  Equal  vectors  possess  the  same  magnitude 
and  the  same  direction.  A  vector  is  not  altered  by  shifting  it 
parallel  to  itself.  A  null  or  zero  vector  is  one  whose  mag- 
nitude is  zero.  To  multiply  a  vector  by  a  positive  scalar 
multiply  its  length  by  that  scalar  and  leave  its  direction 
unchanged.  To  multiply  a  vector  by  a  negative  scalar  mul- 
tiply its  length  by  that  scalar  and  reverse  its  direction. 

Vectors  add  according  to  the  parallelogram  law.  To  subtract 
a  vector  reverse  its  direction  and  add.  Addition,  subtrac- 
tion, and  multiplication  of  vectors  by  a  scalar  follow  the  same 
laws  as  addition,  subtraction,  and  multiplication  in  ordinary- 
algebra.  A  vector  may  be  resolved  into  three  components 
parallel  to  any  three  non-coplanar  vectors.  This  resolution 
can  be  accomplished  in  only  one  way. 

r  =  xa,  +  y'b  +  zc.  (4) 

The  components  of  equal  vectors,  parallel  to  three  given 
non-coplanar  vectors,  are  equal,  and  conversely  if  the  com- 
ponents are  equal  the  vectors  are  equal.  The  three  unit 
vectors  i,  j,  k  form  a  right-handed  rectangular  system.  In 

1  This  limit  exists  and  is  unique.  It  is  independent  of  the  method  in  which 
the  polyhedral  surface  approaches  the  curved  surface. 


52  VECTOR  ANALYSIS 

terms  of  them  any  vector  may  be  expressed  by  means  of  the 
Cartesian  coordinates  x,  y,  z. 

r  =  xi  +  y\  +  zk.  (6) 

Applications.     The  point  which  divides  a  line  in  a  given 
ratio  m  :  n  is  given  by  the  formula 

(7) 


m  +  n 

The  necessary  and  sufficient  condition  that  a  vector  equation 
represent  a  relation  independent  of  the  origin  is  that  the  sum 
of  the  scalar  coefficients  in  the  equation  be  zero.  Between 
any  four  vectors  there  exists  an  equation  with  scalar  coeffi- 
cients. If  the  sum  of  the  coefficients  is  zero  the  vectors  are 
termino-coplanar.  If  an  equation  the  sum  of  whose  scalar 
coefficients  is  zero  exists  between  three  vectors  they  are 
termino-collinear.  The  center  of  gravity  of  a  number  of 
masses  a,  Z>,  c  •  •  •  situated  at  the  termini  of  the  vectors 
A,  B,  C  •  •  •  supposed  to  be  drawn  from  a  common  origin  is 
given  by  the  formula 

.- 


a  +  b  +  c  -\  ---- 

A  vector  may  be  used  to  denote  an  area.  If  the  area  is 
plane  the  magnitude  of  the  vector  is  equal  to  the  magnitude 
of  the  area,  and  the  direction  of  the  vector  is  the  direction  of 
the  normal  upon  the  positive  side  of  the  plane.  The  vector 
representing  a  closed  surface  is  zero. 

EXEECISES  ON  CHAPTER  I 

1.  Demonstrate  the  laws  stated  in  Art.  12. 

2.  A  triangle  may  be  constructed  whose  sides  are  parallel 
and  equal  to  the  medians  of  any  given  triangle. 


ADDITION  AND  SCALAR  MULTIPLICATION          53 

3.  The  six  points  in  which  the  three  diagonals  of  a  com- 
plete quadrangle  1  meet  the  pairs  of  opposite  sides  lie  three 
by  three  upon  four  straight  lines. 

4.  If  two  triangles  are  so  situated  in  space  that  the  three 
points  of  intersection  of  corresponding  sides  lie  on  a  line,  then 
the  lines  joining  the  corresponding  vertices  pass  through  a 
common  point  and  conversely. 

5.  Given  a  quadrilateral  in  space.     Find  the  middle  point 
of  the  line  which  joins  the  middle  points  of  the  diagonals. 
Find  the  middle  point  of  the  line  which  joins  the  middle 
points  of  two  opposite  sides.     Show  that  these  two  points  are 
the  same  and  coincide  with  the  center  of  gravity  of  a  system 
of  equal  masses  placed  at  the  vertices  of  the  quadrilateral. 

6.  If  two  opposite  sides  of  a  quadrilateral  in  space  be 
divided  proportionally  and  if  two  quadrilaterals  be  formed  by 
joining  the  two  points  of  division,  then  the  centers  of  gravity 
of  these  two  quadrilaterals  lie  on  a  line  with  the  center  of 
gravity  of  the  original  quadrilateral.     By  the  center  of  gravity 
is  meant  the  center  of  gravity  of  four  equal  masses  placed  at 
the  vertices.     Can  this  theorem  be  generalized  to  the   case 
where  the  masses  are  not  equal? 

7.  The  bisectors  of  the  angles  of  a  triangle  meet  in  a 
point. 

8.  If  the  edges  of  a  hexahedron  meet  four  by  four  in  three 
points,  the  four  diagonals  of  the  hexahedron  meet  in  a  point. 
In  the  special  case  in  which  the  hexahedron  is  a  parallelepiped 
the  three  points  are  at  an  infinite  distance. 

9.  Prove  that  the  three  straight  lines  through  the  middle 
points  of  the  sides  of  any  face  of  a  tetrahedron,  each  parallel 
to  the  straight  line  connecting  a  fixed  point  P  with  the  mid- 
dle point  of  the  opposite  edge  of  the  tetrahedron,  meet  in  a 

1  A  complete  quadrangle  consists  of  the  six  straight  lines  which  may  be  passed 
through  four  points  no  three  of  which  are  collinear.  The  diagonals  are  the  lines 
which  join  the  points  of  intersection  of  pairs  of  sides. 


54  VECTOR  ANALYSIS 

point  E  and  that  this  point  is  such  that  PE  passes  through 
and  is  bisected  by  the  center  of  gravity  of  the  tetrahedron. 

10.  Show  that  without  exception  there  exists  one  vector 
equation  with   scalar  coefficients  between  any  four  given 
vectors  A,  B,  C,  L. 

11.  Discuss  the  conditions,  imposed  upon  three,  four,  or 
five  vectors  if  they  satisfy  two  equations  the  sum  of  the  co- 
efficients in  each  of  which  is  zero. 


CHAPTER  II 

DIRECT  AND   SKEW  PEODUCTS   OF  VECTORS 

Products  of  Two  Vectors 

27.]  THE  operations  of  addition,  subtraction,  and  scalar 
multiplication  have  been  defined  for  vectors  in  the  way 
suggested  by  physics  and  have  been  employed  in  a  few 
applications.  It  now  becomes  necessary  to  introduce  two 
new  combinations  of  vectors.  These  will  be  called  products 
because  they  obey  the  fundamental  law  of  products ;  i.  e.,  the 
distributive  law  which  states  that  the  product  of  A  into  the 
sum  of  B  and  C  is  equal  to  the  sum  of  the  products  of  A  into 
B  and  A  into  C. 

Definition :  The  direct  product  of  two  vectors  A  and  B  is 
the  scalar  quantity  obtained  by  multiplying  the  product  of 
the  magnitudes  of  the  vectors  by  the  cosine  of  the  angle  be- 
tween them. 

The  direct  product  is  denoted  by  writing  the  two  vectors 

with  a  dot  between  them  as 

A-B. 

This  is  read  A  dot  B  and  therefore  may  often  be  called  the 
dot  product  instead  of  the  direct  product.  It  is  also  called 
the  scalar  product  owing  to  the  fact  that  its  value  is  sca- 
lar. If  A  be  the  magnitude  of  A  and  B  that  of  B,  then  by 

definition 

A-B  =  ^£cos  (A,B).  (1) 

Obviously  the  direct  product  follows  the  commutative  law 
A  •  B  =  B  •  A.  (2) 


56  VECTOR  ANALYSIS 

If  either  vector  be  multiplied  by  a  scalar  the  product  is 
multiplied  by  that  scalar.  That  is 

(x  A)  •  B  =  A  •  (x  B)  =  x  (A  •  B). 

In  case  the  two  vectors  A  and  B  are  collinear  the  angle  be- 
tween them  becomes  zero  or  one  hundred  and  eighty  degrees 
and  its  cosine  is  therefore  equal  to  unity  with  the  positive  or 
negative  sign.  Hence  the  scalar  product  of  two  parallel 
vectors  is  numerically  equal  to  the  product  of  their  lengths. 
The  sign  of  the  product  is  positive  when  the  directions  of  the 
vectors  are  the  same,  negative  when  they  are  opposite.  The 
product  of  a  vector  by  itself  is  therefore  equal  to  the  square 

of  its  length 

A-A=^2.  (3) 

Consequently  if  the  product  of  a  vector  by  itself  vanish  the 
vector  is  a  null  vector. 

In  case  the  two  vectors  A  and  B  are  perpendicular  the 
angle  between  them  becomes  plus  or  minus  ninety  degrees 
and  the  cosine  vanishes.  Hence  the  product  A  •  B  vanishes. 
Conversely  if  the  scalar  product  A  •  B  vanishes,  then 

A  B  cos  (A,  B)  =  0. 

Hence  either  A  or  B  or  cos  (A,  B)  is  zero,  and  either  the 
vectors  are  perpendicular  or  one  of  them  is  null.  Thus  the 
condition  for  the  perpendicularity  of  two  vectors,  neither  of 
which  vanishes,  is  A  •  B  =  0. 

28.]  The  scalar  products  of  the  three  fundamental  unit 
vectors  i,  j,  k  are  evidently 

i.i  =  j.j  =  k-k  =  l,  (4) 


If  more  generally  a  and  b  are  any  two  unit  vectors  the 
product 

a  •  b  =  cos  (a,  b). 


DIRECT  AND  SKEW  PRODUCTS  OF  VECTORS        57 

Thus  the  scalar  product  determines  the  cosine  of  the  angle 
between  two  vectors  and  is  in  a  certain  sense  equivalent  to 
it.  For  this  reason  it  might  be  better  to  give  a  purely 
geometric  definition  of  the  product  rather  than  one  which 
depends  upon  trigonometry.  This  is  easily  accomplished  as 
follows :  If  a  and  b  are  two  unit  vectors,  a  •  b  is  the  length 
of  the  projection  of  either  upon  the  other.  If  more  generally 
A  and  B  are  any  two  vectors  A  •  B  is  the  product  of  the  length 
of  either  by  the  length  of  projection  of  the  other  upon  it. 
From  these  definitions  the  facts  that  the  product  of  a  vector 
by  itself  is  the  square  of  its  length  and  the  product  of  two 
perpendicular  vectors  is  zero  follow  immediately.  The  trigo- 
nometric definition  can  also  readily  be  deduced. 

The  scalar  product  of  two  vectors  will  appear  whenever  the 
cosine  of  the  included  angle  is  of  importance.  The  following 
examples  may  be  cited.  The  projection  of  a  vector  B  upon  a 
vector  A  is 

— '—  A  =  ——  A  a  cos  (A,  B)  =  B  cos  (A,  B)  a,        (5) 
A«  A          A  A 

where  a  is  a  unit  vector  in  the  direction  of  A.  If  A  is  itself  a 
unit  vector  the  formula  reduces  to 

(A  •  B)  A  =  B  cos  (A,  B)  A. 

If  A  be  a  constant  force  and  B  a  displacement  .the  work  done 
by  the  force  A  during  the  displacement  is  A  •  B.  If  A  repre- 
sent a  plane  area  (Art.  25),  and  if  B  be  a 
vector  inclined  to  that  plane,  the  scalar  prod- 
uct A  •  B  will  be  the  volume  of  the  cylinder 
of  which  the  area  A  is  the  base  and  of 
which  B  is  the  directed  slant  height.  For 
the  volume  (Fig.  16)  is  equal  to  the  base  FlG  1 

A   multiplied   by  the    altitude   h.     This   is 
the  projection  of  B  upon  A  or  B  cos  (A,  B).     Hence 

v  =  A  h  =  A  B  cos  (A,  B)  =  A  •  B. 


58  VECTOR  ANALYSIS 

29.]  The  scalar  or  direct  product  follows  the  distributive 
law  of  multiplication.  That  is 

(A  +  B)  •  C  =  A  •  C  +  B  •  C.  (6) 

This  may  be  proved  by  means  of  projections.  Let  C  be  equal 
to  its  magnitude  C  multiplied  by  a  unit  vector  c  in  its  direc- 
tion. To  show 

(A  +  B)  •  (G  c)  =  A  •  (tfc)  +  B.  (Co) 
or  (A  +  B)  •  c  =  A  •  c  +  B  •  c. 

A  •  c  is  the  projection  of  A  upon  c  ;  B  •  c,  that  of  B  upon  c  ; 
(A  +  B)  •  c,  that  of  A  +  B  upon  c.  But  the  projection  of  the 
sum  A  +  B  is  equal  to  the  sum  of  the  projections.  Hence 
the  relation  (6)  is  proved.  By  an  immediate  generalization 

(A  +  B+...)-(P  +  ft+---)  =  A.P  +  A.a+... 

+  B.P  +  B.Q  +  ...    e 


The  scalar  product  may  be  used  just  as  the  product  in  ordi- 
nary algebra.     It  has  no  peculiar  difficulties. 

If  two  vectors  A  and  B  are  expressed  in  terms  of  the 
three  unit  vectors  i,  j,  k  as 

A  =  A1  i  +  Az  j  +  As  k, 

and  B  =  Bl  i  +  #2  j  +  Bz  k, 

then        A.B^^i  +  ^j  +  ^gk)  .(^1  +  £2j  +  £3k) 
=  Al  B^  i  •  i  +  Al  £2  i  •  j  +  Al  Bz  i  .  k 
+  A  2  Bl  j  •  i  +  A  2  £2  j  •  j  +  A  2  #,  j  .  k 
+  ^3£1k.j  +  ,43£2k.j  +  ^3£8k.k. 

By  means  of  (4)  this  reduces  to 

A  •  B  =  Al  Bl  +  4,  £2  +  A8  JSS.  (7) 

If  in  particular  A  and  B  are  unit  vectors,  their  components 
A1^AZ^A8  and  JS^Szt£&  are  the  direction  cosines  of  the 
lines  A  and  B  referred  to  X,  Y,  Z. 


DIRECT  AND  SKEW  PRODUCTS  OF  VECTORS 


59 


A1  =  cos  (A,  JT),    Az  =  cos  (A,  F),    AB  =  cos  (A,  £"), 
^  =  cos  (B,  JT),     £3  =  cos  (B,  F),     ^3  =  cos  (B»  <&)• 

Moreover  A  •  B  is  the  cosine  of  the  included  angle.     Hence 
the  equation  becomes 

cos  (A,  B)  =  cos  (A,  Z)  cos  (B,  Z)  +  cos  (A,  F)  cos  (B,  F) 

+  cos  (A,£)  cos  (B,^). 

In  case  A  and  B  are  perpendicular  this  reduces  to  the  well- 
known  relation 

0  =  cos  (A,  Z)  cos  (B,  Z)  +  cos  (A,  F)  cos  (B,  F) 

+  cos    A,£    cos    B^ 


between  the  direction  cosines  of  the 
line  A  and  the  line  B. 

30.]  If  A  and  B  are  two  sides  0  A 
and  OB  of  a  triangle  OAB,  the  third 
side  ^£isC  =  B-A  (Fig.  17). 

C  •  C  =  (B  -  A)  •  (B  -  A)  =  B  •  B  +  A  •  A  -  2  A  •  B  cos  (A,  B)  , 
or  C*  =  A*  +  £*-2A£cos  (AB). 

That  is,  the  square  of  one  side  of  a  triangle  is  equal  to  the 
sum  of  the  squares  of  the  other  two  sides  diminished  by  twice 
their  product  times  the  cosine  of  the  angle  between  them. 
Or,  the  square  of  one  side  of  a  triangle  is  equal  to  the  sum  of 
the  squares  of  the  other  two  sides  diminished  bj  twice  the 
projection  of  either  of  those  sides  upon  the  other,  the  theorem 
sometimes  known  as  the  generalized  Pythagorean  theorem. 

If  A  and  B  are  two  sides  of  a  parallelogram,  C  =  A  +  B 
and  D  =  A  —  B  are  the  diagonals.     Then 

C.C  =  (A  +  B).(A  +  B)=A.A  +  2A.B  +  B«B, 
D«D  =  (A-B).(A-B)=A.A-2A-B  +  B.B, 


or 


=  2 


60  VECTOR  ANALYSIS 

That  is,  the  sum  of  the  squares  of  the  diagonals  of  a  parallelo- 
gram is  equal  to  twice  the  sum  of  the  squares  of  two  sides. 
In  like  manner  also 

C-C-D.D  =  4  A-B 

or  C*-D*  =  ±AB  cos  (A,  B). 

That  is,  the  difference  of  the  squares  of  the  diagonals  of  a 
parallelogram  is  equal  to  four  times  the  product  of  one  of  the 
sides  by  the  projection  of  the  other  upon  it. 

If  A  is  any  vector  expressed  in  terms  of  i,  j,  k  as 

A  =  Al  i  -f  A2  j  +  AB  k, 
then  A  •  A  =  A2  =  A?  +  A*  +  A*.  (8) 

But  if  A  be  expressed  in  terms  of  any  three  non-coplanar  unit 
vectors  a,  b,  c  as 


A2  =  a?  +  ft2  +  c2  +  2  a  b  cos    a,  b)  +  2  b  c  cos  (b,  c) 

+  2  c  a  cos  (c,  a)  . 

This  formula  is  analogous  to  the  one  in  Cartesian  geometry 
which  gives  the  distance  between  two  points  referred  to 
oblique  axes.  If  the  points  be  x^  yv  zv  and  xv  yz,  zz  the 
distance  squared  is 

D2  =  (a,  -  zj)2  +  (y,  -  yi)«  +  (*,  -  Zi)2 
+  2  O2  -  xj  (2/2  -  yj)  cos  (JT,  F) 

+  2  (2/2  -  2/i)  («a  ~  *i)  cos  (^  ^) 
+  2  02  -zj  (xz  -  ajj)  cos  (^,X). 

31.]  Definition:  The  skew  product  of  the  vector  A  into 
the  vector  B  is  the  vector  quantity  C  whose  direction  is  the 
normal  upon  that  side  of  the  plane  of  A  and  B  on  which 


DIRECT  AND  SKEW  PRODUCTS   OF  VECTORS        61 

rotation  from  A  to  B  through  an  angle  of  less  than  one 
hundred  and  eighty  degrees  appears  positive  or  counter- 
clockwise ;  and  whose  magnitude  is  obtained  by  multiplying 
the  product  of  the  magnitudes  of  A  and  B  by  the  sine  of  the 
angle  from  A  to  B. 

The   direction   of  A  X  B  may  also  be  denned  as  that  in 
which    an    ordinary   right-handed 


AXB 
B* 


screw  advances  as  it  turns  so  as 
to  carry  A  toward  B  (Fig.  18). 

The  skew  product  is  denoted  by 
a  cross  as  the  direct  product  was 
by  a  dot.  It  is  written  FIG.  18. 

C  =  AxB 

and  read  A  cross  B.  For  this  reason  it  is  often  called  the  cross 
product.  More  frequently,  however,  it  is  called  the  vector  prod- 
uct, owing  to  the  fact  that  it  is  a  vector  quantity  and  in  con- 
trast with  the  direct  or  scalar  product  whose  value  is  scalar. 
The  vector  product  is  by  definition 

C  =  A  X  B  =  A  B  sin  (A,  B)  c,  (9) 

when  A  and  B  are  the  magnitudes  of  A  and  B  respectively  and 
where  c  is  a  unit  vector  in  the  direction  of  C.  In  case  A  and 
B  are  unit  vectors  the  skew  product  A  X  B  reduces  to  the 
unit  vector  c  multiplied  by  the  sine  of  the  angle  from  A  to  B. 
Obviously  also  if  either  vector  A  or  B  is  multiplied  by  a  scalar 
x  their  product  is  multiplied  by  that  scalar. 

(x  A)  x  B  =  A  x  (zB)  =  xC. 

If  A  and  B  are  parallel  the  angle  between  them  is  either  zero 
or  one  hundred  and  eighty  degrees.  In  either  case  the  sine 
vanishes  and  consequently  the  vector  product  A  X  B  is  a  null 
vector.  And  conversely  if  A  X  B  is  zero 

A  B  sin  (A,  B)  =  0. 


^62  VECTOR  ANALYSIS 

Hence  A  or  B  or  sin  (A,  B)  is  zero.  Thus  the  condition  for 
parallelism  of  two  vectors  neither  of  which  vanishes  is  A  X  B 
=  0.  As  a  corollary  the  vector  product  of  any  vector  into 
itself  vanishes. 

32.]  The  vector  product  of  two  vectors  will  appear  wher- 
ever the  sine  of  the  included  angle  is  of  importance,  just  as 
the  scalar  product  did  in  the  case  of  the  cosine.  The  two  prod- 
ucts are  in  a  certain  sense  complementary.  They  have  been 
denoted  by  the  two  common  signs  of  multiplication,  the  dot 
and  the  cross.  In  vector  analysis  they  occupy  the  place  held 
by  the  trigonometric  functions  of  scalar  analysis.  They  are 
at  the  same  time  amenable  to  algebraic  treatment,  as  will  be 
seen  later.  At  present  a  few  uses  of  the  vector  product  may 
be  cited. 

If  A  and  B  (Fig.  18)  are  the  two  adjacent  sides  of  a  parallel- 
ogram the  vector  product 

C  =  A  x  B  =  A  B  sin  (A,  B)  c 

represents  the  area  of  that  parallelogram  in  magnitude  and 
direction  (Art.  25).  This  geometric  representation  of  A  x  B 
is  of  such  common  occurrence  and  importance  that  it  might 
well  be  taken  as  the  definition  of  the  product.  From  it  the 
trigonometric  definition  follows  at  once.  The  vector  product 
appears  in  mechanics  in  connection  with  couples.  If  A  and 
—  A  are  two  forces  forming  a  couple,  the  moment  of  the 
couple  is  A  x  B  provided  only  that  B  is  a  vector  drawn  from 
any  point  of  A  to  any  point  of  —  A.  The  product  makes  its 
appearance  again  in  considering  the  velocities  of  the  individ- 
ual particles  of  a  body  which  is  rotating  with  an  angular  ve- 
locity given  in  magnitude  and  direction  by  A.  If  B,  be  the 
radius  vector  drawn  from  any  point  of  the  axis  of  rotation  A 
the  product  A  x  E  will  give  the  velocity  of  the  extremity  of 
B  (Art.  51).  This  velocity  is  perpendicular  alike  to  the  axis 
of  rotation  and  to  the  radius  vector  R. 


DIRECT  AND  SKEW  PRODUCTS  OF   VECTORS        63 

33.]  The  vector  products  A  X  B  and  B  x  A  are  not  the 
same.  They  are  in  fact  the  negatives  of  each  other.  For  if 
rotation  from  A  to  B  appear  positive  on  one  side  of  the  plane 
of  A  and  B,  rotation  from  B  to  A  will  appear  positive  on  the 
other.  Hence  A  x  B  is  the  normal  to  the  plane  of  A  and  B 
upon  that  side  opposite  to  the  one  upon  which  B  x  A  is  the 
normal.  The  magnitudes  of  A  X  B  and  B  X  A  are  the  same. 

Hence 

AxB  =  -BxA.  (10) 

The  factors  in  a  vector  product  can  be  interchanged  if  and  only 
if  the  sign  of  the  product  be  reversed. 

This  is  the  first  instance  in  which  the  laws  of  operation  in 
vector  analysis  differ  essentially  from  those  of  scalar  analy- 
sis. It  may  be  that  at  first  this  change  of  sign  which  must 
accompany  the  interchange  of  factors  in  a  vector  product  will 
give  rise  to  some  difficulty  and  confusion.  Changes  similar  to 
this  are,  however,  very  familiar.  No  one  would  think  of  inter- 
changing the  order  of  x  and  y  in  the  expression  sin  (x  —  y) 
without  prefixing  the  negative  sign  to  the  result.  Thus 

sin  (y  —  x)  =  —  sin  (x  —  y), 

although  the  sign  is  not  required  for  the  case  of  the  cosine, 
cos  (y  —  x)  =  cos  (  x  —  y~). 

Again  if  the  cyclic  order  of  the  letters  A  B  C  in  the  area  of  a 
triangle  be  changed,  the  area  will  be  changed  in  sign  (Art. 

25). 

A  B  C  =  -A  CB. 

In  the  same  manner  this  reversal  of  sign,  which  occurs 
when  the  order  of  the  factors  in  a  vector  product  is  reversed, 
will  appear  after  a  little  practice  and  acquaintance  just  as 
natural  and  convenient  as  it  is  necessary. 

34.]  The  distributive  law  of  multiplication  holds  in  the 
case  of  vector  products  just  as  in  ordinary  algebra  —  except 


64 


VECTOR   ANALYSIS 


A 

FIG.  19. 


that  the   order  of  the  factors  must    be   carefully   maintained 
when  expanding. 

A  very  simple  proof  may  be  given  by  making  use  of  the  ideas 
developed  in  Art.  26.  Suppose  that  C 
is  not  coplanar  with  A  and  B.  Let  A 
and  B  be  two  sides  of  a  triangle  taken 
in  order.  Then  —  (A  +  B)  will  be  the 
third  side  (Fig.  19).  Form  the  prism 
of  which  this  triangle  is  the  base  and 
of  which  C  is  the  slant  height  or  edge. 
The  areas  of  the  lateral  faces  of  this 
prism  are 

A  X  C,     B  X  C,     -  (A  +  B)  X  C. 
The  areas  of  the  bases  are 

s  (A  X  B)  and  —  5  (A  x  B). 

ii  —  • 

But  the  sum  of  all  the  faces  of  the  prism  is  zero;  for  the 
prism  is  a  closed  surface.  Hence 

A  x  C  +  B  x~C  -  (A  +  B)  X  C  +  \  (A  X  B)  -  \  (A  X  B)  =  0, 

AxC  +  BxC  —  (A  +  B)  x  C  =  0, 
or  A  x  C  +  B  x  C  =  (A  4-  B)  x  C.  (11) 

The  relation  is  therefore  proved  in  case  C  is  non-coplanar 
with  A  and  B.  Should  C  be  coplanar  with  A  and  B,  choose  D, 
any  vector  out  of  that  plane.  Then  C  +  D  also  will  lie  out  of 
that  plane.  Hence  by  (11) 

A  x  (C  +  D)  +  B  x  (C  +  D)  =  (A  +  B)  x  (C  +  D). 

Since  the  three  vectors  in  each  set  A,  C,  D,  and  B,  C,  D,  and 
A  +  B,  C,  D  will  be  non-coplanar  if  D  is  properly  chosen,  the 
products  may  be  expanded. 


DIRECT   AND  SKEW  PRODUCTS   OF  VECTORS       65 

AxC+AxD+BxC+BxD 

=  (A  +  B)  x  C  +  (A  +  B)  x  D. 
But  by  (11)     A^-D  +  BxD=(A  +  B)xD. 
Hence  AxC  +  BxC—  (A  +  B)xC. 

This  completes  the  demonstration.     The  distributive  law  holds 
for  a  vector  product.     The  generalization  is  immediate. 

(A  +  B  +  •  •  •)  x  (P  +  <J  +  •  •  •)  =  AxP  +  Axft+---     (11)' 

+  B  x  P  +  B  X  Q  +  ••• 


35.]  The  vector  products  of  the  three  unit  vectors  i,  j,  k  are 
easily  seen  by  means  of  Art.  17  to  be 

ixi  =  jxj  =  kxk  =  0, 

i  x  j  =  —  j  x  i  =  k,  (12) 

j  xk  —  —  k  x  j  =  i, 
k  x  i  =  —  ixk—  j. 

The  skew  product  of  two  equal  1  vectors  of  the  system  i,  j,  k 
is  zero.  The  product  of  two  unequal  vectors  is  the  third  taken 
with  the  positive  sign  if  the  vectors  follow  in  the  cyclic  order 
i  j  k  but  with  the  negative  sign  if  they  do  not. 

If  two  vectors  A  and  B  are  expressed  in  terms  of  i,  j,  k, 
their  vector  product  may  be  found  by  expanding  according 
to  the  distributive  law  and  substituting. 


vi4->4     7?   i  v  i  -4-   >4    7?    i  v  k 

^.     1.       |^     -*X  -I    -i-'o  A     /'X    J        K     -^*-1    •*-'    O  *     -^      *^ 
1  £t  •!        '  I  O 

,i     j4     T)  i  y  i  -4-  >4    ^  i  v  i    i     >4     ??    i  y  lr 

+  A3  Bl  k  x  i  +  A3  BZ  k  x  j  4-  -43  J?3  k  x  k. 
Hence     A  x  B  =  (^42  ^3  —  A3  BJ)  i  +  (A3  B^  —  A}  ^3)  J 

+  (A1BZ-A,B^*. 

1  This  follows  also  from  the  fact  that  the  sign  is  changed  when  the  order  of 
factors  is  reversed.    Hence  jXj  =  —  j  X  j  =  0. 

6  , 


•  MifliU     «-*V*r*    -/ 

s 


66 


VECTOR  ANALYSIS 


This  may  be  written  in  the  form  of  a  determinant  as 


A  x  B  = 


Ac 


The  formulas  for  the  sine  and  cosine  of  the  sum  or  dif- 
ference of  two  angles  follow  immediately  from  the  dot  and 
cross  products.  Let  a  and  b  be  two  unit  vectors  lying  in  the 
i  j -plane.  If  x  be  the  angle  that  a  makes  with  i,  and  y  the 
angle  b  makes  with  i,  then 


Hence 
If 

Hence 


Hence 


Hence 


a 
b 

a  •  b 
a-b 

cos  (y  —  x) 
b' 

a-b' 

cos  (y  +  x) 
a  x  b 
a  x  b 

sin  (y  —  x)  •• 
a  x  b' 
a  x  b' 
sin  (y  +  x) 


cos  x  i  +  sin  x  j," 
cos  y  i  +  sin  y  j, 
cos  (a,  b)  =  cos  (y  —  x), 
cos  x  cos  y  +  sin  x  sin  y. 
cos  y  cos  x  +  sin  y  sin  x. 
cos  y  i  —  sin  y  j, 
cos  (a,  b')  =  cos  (y  +  x). 
cos  y  cos  x  —  sin  y  sin  x. 
k  sin  (a,  b)  =  k  sin  {y  —  #), 
k  (sin  y  cos  x  —  sin  x  cos  y). 
sin  y  cos  x  —  sin  a;  cos  y. 
k  sin  (a,  b')  =  k  sin  (y  +  #) 
k  (sin  y  cos  x  +  sin  x  cos  ?/). 
sin  ^  cos  #  +  sin  x  cos  y. 


If  Z,  m,  w  and  l\im\n1  are  the  direction  cosines  of  two 
unit  vectors  a  and  a'  referred  to  JL,  F,  Z,  then 


a  •  a'  =  cos  (a,  a')  = 


w'k, 

+  m  m'  + 


as  has  already  been  shown  in  Art.  29.  The  familiar  formula 
for  the  square  of  the  sine  of  the  angle  between  a  and  a'  may 
be  found. 


DIRECT  AND  SKEW  PRODUCTS  OF  VECTORS        67 

a  x  a'  =  sin  (a,  ar)  e  =  (run'  —  m'  n)  i  +  (nl'  —  n'  I)  j 
-f  (7m'  —  I'm)  k, 

where  e  is  a  unit  vector  perpendicular  to  a  and  af. 

(a  x  a')  •  (a  X  a')  =  sin2  (a,  a')  e  •  e  =  sin2  (a,  a'). 
sin2(a,a')  =  (mn'-m'n)z+  (nl'-  n'  Z)2  +(lm'  -  I'  w)2. 

This  leads  to  an  easy  way  of  establishing  the  useful  identity 


-  (7 


Products  of  More  than  Two  Vectors 

36.]  Up  to  this  point  nothing  has  been  said  concerning 
products  in  which  the  number  of  vectors  is  greater  than 
two.  If  three  vectors  are  combined  into  a  product  the  result 
is  called  a  triple  product.  Next  to  the  simple  products 
A«B  and  AxB  the  triple  products  are  the  most  important. 
All  higher  products  may  be  reduced  to  them. 

The  simplest  triple  product  is  formed  by  multiplying  the 
scalar  product  of  two  vectors  A  and  B  into  a  third  C  as 

(A-B)  C. 

This  in  reality  does  not  differ  essentially  from  scalar  multi- 
plication (Art.  6).  The  scalar  in  this  case  merely  happens  to 
be  the  scalar  product  of  the  two  vectors  A  and  B.  Moreover 
inasmuch  as  two  vectors  cannot  stand  side  by  side  in  the 
form  of  a  product  as  B  C  without  either  a  dot  or  a  cross  to 
unite  them,  the  parenthesis  in  (A-B)  C  is  superfluous.  The 
expression  .  «  „ 

cannot  be  interpreted  in  any  other  way  l  than  as  the  product 
of  the  vector  C  by  the  scalar  A»B. 

i  Later  (Chap.  V.)  the  product  BC,  where  no  sign  either  dot  or  cross  occurs, 
will  be  defined.  But  it  will  be  seen  there  that  (A.B)  C  and  A»(BC)  are  identical 
and  consequently  no  ambiguity  can  arise  from  the  omission  of  the  parenthesis. 


68  VECTOR  ANALYSIS 

37.]     The  second  triple  product  is  the  scalar  product  of 
two  vectors,  of  which  one  is  itself  a  vector  product,  as 
A»(BxC)or  (AxB>C. 

This  sort  of  product  has  a  scalar  value  and  consequently  is 

often  called  the  scalar  triple  prod- 
uct. Its  properties  are  perhaps  most 
easily  deduced  from  its  commonest 
geometrical  interpretation.  Let  A,  B, 
and  C  be  any  three  vectors  drawn 
from  the  same  origin  (Fig.  20). 
Then  BxC  is  the  area  of  the  par- 
allelogram of  which  B  and  C  are  two  adjacent  sides.  The 

scalar  A-(BxC)  =  »  (14) 

will  therefore  be  the  volume  of  the  parallelepiped  of  which 
BxC  is  the  base  and  A  the  slant  height  or  edge.  See  Art.  28. 
This  volume  v  is  positive  if  A  and  BxC  lie  upon  the  same 
side  of  the  B  C-plane ;  but  negative  if  they  lie  on  opposite 
sides.  In  other  words  if  A,  B,  C  form  a  right-handed  or 
positive  system  of  three  vectors  the  scalar  A«(BxC)  is  posi- 
tive; but  if  they  form  a  left-handed  or  negative  system,  it 
is  negative. 

In  case  A,  B,  and  C  are  coplanar  this  volume  will  be 
neither  positive  nor  negative  but  zero.  And  conversely  if 
the  volume  is  zero  the  three  edges  A,  B,  C  of  the  parallele- 
piped must  lie  in  one  plane.  Hence  the  necessary  and  suffi- 
cient condition  for  the  coplanarity  of  three  vectors  A,  B,  C  none 
of  which  vanishes  is  A»(BxC)  =  0.  As  a  corollary  the  scalar 
triple  product  of  three  vectors  of  which  two  are  equal  or 
collinear  must  vanish ;  for  any  two  vectors  are  coplanar. 

The  two  products  A»(BxC)  and  (AxB)«C  are  equal  to  the 
same  volume  v  of  the  parallelepiped  whose  concurrent  edges 
are  A,  B,  C.  The  sign  of  the  volume  is  the  same  in  both 

cases.     Hence 

(AxB)-C  =  A.(BxC)  =  v.  (14)' 


DIRECT  AND  SKEW  PRODUCTS  Of   VECTORS        69 

This  equality  may  be  stated  as  a  rule  of  operation.  The  dot 
and  the  cross  in  a  scalar  triple  product  may  ~be  interchanged 
without  altering  the  value  of  the  product. 

It  may  also  be  seen  that  the  vectors  A,  B,  C  may  be  per- 
muted cyclicly  without  altering  the  product. 

A-(BxC)  =  B.(CxA)  =  C.(AxB).  (15) 

For  each  of  the  expressions  gives  the  volume  of  the  same 
parallelepiped  and  that  volume  will  have  in  each  case  the 
same  sign,  because  if  A  is  upon  the  positive  side  of  the  B  C- 
plane,  B  will  be  on  the  positive  side  of  the  C  A-plane  and  C 
upon  the  positive  side  of  the  A  B-plane.  The  triple  product 
may  therefore  have  any  one  of  six  equivalent  forms 

A-(BxC)  =  B-(CxA)  =  i-(AxB)  (15)' 

=  (AxB>C  =  (BxC).A  =  (CxA)-B. 

If  however  the  cyclic  order  of  the  letters  is  changed  the 
product  will  change  sign. 

A-(BxC)  =  -  A<CxB).  (16) 

This  may  be  seen  from  the  figure  or  from  the  fact  that 
BxC  =  —  CxB. 

Hence :  A  scalar  triple  product  is  not  altered  by  interchanging 
the  dot  or  the  cross  or  by  permuting  cyclicly  the  order  of  the 
vectors,  but  it  is  reversed  in  sign  if  the  cyclic  order  be  changed. 

38.]  A  word  is  necessary  upon  the  subject  of  parentheses 
in  this  triple  product.  Can  they  be  omitted  without  am- 
biguity ?  They  can.  The  expression 

A-BxC 

can  have  only  the  one  interpretation 

A<BxC). 

For  the  expression  (A«B)xC  is  meaningless.  It  is  impos- 
sible to  form  the  skew  product  of  a  scalar  A-B  and  a  vector 


70  VECTOR  ANALYSIS 

C.  Hence  as  there  is  only  one  way  in  which  A«BxC  may 
be  interpreted,  no  confusion  can  arise  from  omitting  the 
parentheses.  Furthermore  owing  to  the  fact  that  there  are 
six  scalar  triple  products  of  A,  B,  and  C  which  have  the  same 
value  and  are  consequently  generally  not  worth  distinguish- 
ing the  one  from  another,  it  is  often  convenient  to  use  the 

symbol 

[ABC] 

to  denote  any  one  of  the  six  equal  products. 

[A  B  C]  =  A-BxC  =  B-CxA  =  C-AxB 
=  AxB.C  =  BxC-A  ='  CxA-B 
then  [A  B  C]  =  -  [A  C  B].  (16)' 

The  scalar  triple  products  of  the  three  unit  vectors  i,  j,  k 
all  vanish  except  the  two  which  contain  the  three  different 

vectors. 

[ijk]=-[ikj]  =  l.  (17) 

Hence  if  three  vectors  A,  B,  C  be  expressed  in  terms  of  i,  j,  k 

as 

A  =  Al  i  +  Az  j  +  A3  k, 

B  =  Bl  i  +  £2  j  +  #,  k, 
C=Cli+  <72j  +  C3k, 

then      [A  B  C]  =  Al  £z  C3  +  Bl  C2  A3  +  C,  At  Bz 
-  A,  Bz  <72  -  S1  C3  A2  -  0,  As  Br 

This  may  be  obtained  by  actually  performing  the  multiplica- 
tions which  are  indicated  in  the  triple  product.  The  result 
may  be  written  in  the  form  of  a  determinant.1 

A    A    A 
-™-\  -^2  "* 


[A  B  C]  = 


C1 


(18)' 


1  This  is  the  formula  given  in  solid  analytic  geometry  for  the  volume  of  a 
tetrahedron  one  of  whose  vertices  is  at  the  origin.  For  a  more  general  formula 
see  exercises. 


DIRECT  AND  SKEW  PRODUCTS   OF   VECTORS       71 

If  more  generally  A,  B,  C  are  expressed  in  terms  of  any  three 
non-coplanar  vectors  a,  b,  c  which  are  not  necessarily  unit 

vectors, 

A  =  &J  a  +  «2  b  +  «3  c 

B  =  \  a  +  &2  b  +  &3  c 
C  =  cx  a  +  c2  b  +  c3  c 

where  av  «2,  «3;  bv  62,  J3;  and  cx,  e?2,  c3  are  certain  con- 
stants, then 

[ABC]-  (a1  &2  c3  +  ft,  c2  «3  +  Cl  «2  J3 


a2  —  Cj  a3  62)  [a  b  c]. 


or  [ABC]- 


&1    &2    &l 
Ci     Co    d. 


[a  be]  (19)' 


39.]     The  third  type  of  triple  product  is  the  vector  product 
of  two  vectors  of  which  one  is  itself  a  vector  product.     Such 

are 

Ax(BxC)  and  (AxB)xC. 

The  vector  Ax(BxC)  is  perpendicular  to  A  and  to  (BxC). 
But  (BxC)  is  perpendicular  to  the  plane  of  B  and  C.  Hence 
Ax  (BxC),  being  perpendicular  to  (BxC)  must  lie  in  the 
plane  of  B  and  C  and  thus  take  the  form 

Ax(BxC)  =  x  B  +  y  C, 

where  x  and  y  are  two  scalars.  In  like  manner  also  the 
vector  (AxB)xC,  being  perpendicular  to  (AxB)  must  lie 
in  the  plane  of  A  and  B.  Hence  it  will  be  of  the  form 

(AxB)xC  =  mA  +  wB 

where  m  and  v  are  two  scalars.     From  this  it  is  evident  that 

in  general 

(AxB)xC  is  not  equal  to  Ax  (BxC). 

The  parentheses  therefore  cannot  be  removed  or  inter- 
changed. It  is  essential  to  know  which  cross  product  is 


72  VECTOR  ANALYSIS 

formed  first  and  which  second.     This  product  is  termed  the 
vector  triple  product  in  contrast  to  the  scalar  triple  product. 

The  vector  triple  product  may  be  used  to  express  that  com- 
ponent of  a  vector  B  which  is  perpendicular  to  a  given  vector 
A.  This  geometric  use  of  the  product  is  valuable  not  only  in 

itself  but  for   the  light  it   sheds 


AXB 

AXB 


B 


upon  the  properties  of  the  product. 
Let  A  (Fig.  21)  be  a  given  vector 
and  B  another  vector  whose  com- 
ponents parallel  and  perpendicular 
to  A  are  to  be  found.  Let  the 
components  of  U  parallel  and  per- 
A  X  (AXB)  pendicular  to  A  be  B'and  B"  re- 

p      21  spectively.     Draw  A  and  B  from  a 

common  origin.    The  product  AxB 

is  perpendicular  to  the  plane  of  A  and  B.  The  product 
Ax  (AxB)  lies  in  the  plane  of  A  and  B.  It  is  furthermore 
perpendicular  to  A.  Hence  it  is  collinear  with  B".  An 
examination  of  the  figure  will  show  that  the  direction  of 
Ax  (AxB)  is  opposite  to  that  of  B".  Hence 


where  c  is  some  scalar  constant. 

Now  Ax(AxB)  =~AZB  sin  (A,B)  b" 

but  -  c  B"  =  -  c  B  sin  (A,  B)  b", 

if  b"  be  a  unit  vector  in  the  direction  of  B". 
Hence  c  =  Az  =  A«A. 

Ax  (AxB) 
Hence  B"  = 7-7—  (20) 

AfA 

The  component  of  B  perpendicular  to  A  has  been  expressed 
in  terms  of  the  vector  triple  product  of  A,  A,  and  B.  The 
component  B'  parallel  to  A  was  found  in  Art.  28  to  be 


DIRECT  AND  SKEW  PRODUCTS  OF  VECTORS       73 
B'=£?A  (21) 

00 


40.]     The  vector  triple  product  Ax  (BxC)  may  be  expressed 
as  the  sum  of  two  terms  as 

AX  (BXC)  =  A.C  B-A-B  c 

In  the  first  place  consider  the   product  when   two  of  the 
vectors  are  the  same.     By  equation  (22) 

A-  A  B  =  A-B  A  -  Ax(AxB)  (22) 

or  Ax(AxB)  =  A.B  A  -  A-  A  B  (23) 

This  proves  the  formula  in  case  two  vectors  are  the  same. 
To  prove  it  in  general  express  A  in  terms  of  the  three 
non-coplanar  vectors  B,  C,  and  BxC. 

A  =  6B+cC  +  «  (BxC),  (I) 

where  a,  Z>,  c  are  scalar  constants.     Then 

Ax(BxC)  =  SBx(BxC)  +  cCx(BxC)  (II) 

+  a  (BxC)x(BxC). 

The  vector  product  of  any  vector  by  itself  is  zero.     Hence 

(BxC)x(BxC)  =  0 

Ax(BxC)  =  JBx(BxC)  +  c  Cx(BxC).          (II)' 
By  (23)  Bx(BxC)  =  B-C  B  -  B«B  C 

Cx(BxC)  =  -  Cx(CxB)  =  -  C.B  C  +  C«C  B. 
Hence  Ax(BxC)  =  [(6B-C  +  cC«C)B-  (6B.B  +  cC-B)C].  (II)" 
But  from  (I)     A-B  =  JB-B  +  cC-B  +  a  (BxC)-B 
and  A-C  =  b  B-C  +  c  C»C  +  a  (BxC>C. 

By  Art.  37          (BxC)-B  =  0  and  (BxC>C  =  0. 
Hence  A«B  =  JB»B  +  cC-B, 

A-C  =  &B-C  +  cC-C. 


74  VECTOR  ANALYSIS 

Substituting  these  values  in  (II)", 

Ax(BxC)  =  A-C  B  -  A.B  C.  (24) 

The  relation  is  therefore  proved  for  any  three  vectors  A,  B,  C. 

Another  method  of  giving  the  demonstration  is  as  follows. 

It  was  shown  that  the  vector  triple  product  Ax  (BxC)  was 

of  the  form 

Ax  (BxC)  =  #B  +  2/C. 

Since  Ax(AxC)  is  perpendicular  to  A,  the  direct  product  of 
it  by  A  is  zero.  Hence 

A.[Ax(BxC)]  =  zA-B  +  2/A-C  =  0 
and  x  :  y  =  A«C  .•  —  A-B. 

Hence  Ax  (BxC).  =  n  (A-C  B  —  A«B  C),  .'• 

where  n  is  a  scalar  constant.  It  remains  to  show  n  —  1. 
Multiply  by  B. 

Ax(BxC>B  =  n  (A-C  B-B  -  A-B  C-B). 

The  scalar  triple  product  allows  an  interchange  of  dot  and 
cross.  Hence 

Ax(BxC>B  =  A<BxC)xB  =  -  A-[Bx(BxC)], 

if  the  order  of  the  factors  (BxC)  and  B  be  inverted. 
-A.[Bx(BxC)]  =-A.[B-C  B-B-BC] 

=  -B.C  A,B  +  B.B  A.C. 

Hence  n  =  1  and          Ax(BxC)  =  A-C  B  -  A^B  C.         (24) 

From  tne  three  letters  A,  B,  C  by  different  arrangements, 
four  allied  products  in  each  of  which  B  and  C  are  included  in 
parentheses  may  be  formed.  These  are 

Ax(BxC),     Ax(CxB),     (CxB)xA,     (BxC)xA. 

As  a  vector  product  changes  its  sign  whenever  the  order  of 
two  factors  is  interchanged,  the  above  products  evidently 
satisfy  the  equations 

Ax  (BxC)  =-Ax(CxB)  =  (CxB)xA  =  -(BxC)xA. 


DIRECT  AND  SKEW  PRODUCTS   OF   VECTORS        75 

The  expansion  for  a  vector  triple  product  in  which  the 
parenthesis  comes  first  may  therefore  be  obtained  directly 
from  that  already  found  when  the  parenthesis  comes  last 

(AxB)xC  =  -  Cx(AxB)  =  -  C.B  A  +  C«A  B. 
The  formulae  then  become 

Ax  (BxC)  =  A.C  B  -  A.B  C  (24) 

and  (AxB)xC  =  A-C  B  -  C-B  A.  (24)' 

These  reduction  formulae  are  of  such  constant  occurrence  and 
great  importance  that  they  should  be  committed  to  memory. 
Their  content  may  be  stated  in  the  following  rule.  To  expand 
a  vector  triple  product  first  multiply  the  exterior  factor  into  the 
remoter  term  in  the  parenthesis  to  form  a  scalar  coefficient  for 
the  nearer  one,  then  multiply  the  exterior  factor  into  the  nearer 
term  in  the  parenthesis  to  form  a  scalar  coefficient  for  the 
remoter  one,  and  subtract  this  result  from  the  first. 

41.]  As  far  as  the  practical  applications  of  vector  analysis 
are  concerned,  one  can  generally  get  along  without  any 
formulae  more  complicated  than  that  for  the  vector  triple 
product.  But  it  is  frequently  more  convenient  to  have  at 
hand  other  reduction  formulae  of  which  all  may  be  derived 
simply  by  making  use  of  the  expansion  for  the  triple  product 
Ax  (BxC)  and  of  the  rules  of  operation  with  the  triple  pro- 
duct A»BxC. 

To  reduce  a  scalar  product  of  two  vectors  each  of  which 
is  itself  a  vector  product  of  two  vectors,  as 

(AxB).(CxD). 

Let  this  be  regarded  as  a  scalar  triple  product  of  the  three 
vectors  A,  B,  and  CxD  —  thus 

AxB.(CxD). 

Interchange  the  dot  and  the  cross. 


76  VECTOR  ANALYSIS 

AxB.(CxD)=A-Bx(CxD) 
Bx(CxD)  =  B-D  C  -  B-C  D. 

Hence  (AxB).(CxD)  =  A-C  B-D  -  A-D  B.C.         (25) 

This  may  be  written  in  determinantal  form. 

A-C       A-D 


(AxB)-(CxD)  = 


(25)' 


B-C       B-D 

If  A  and  D  be  called  the  extremes  ;  B  and  C  the  means  ;  A 
and  C  the  antecedents;  B  and  D  the  consequents  in  this 
product  according  to  the  familiar  usage  in  proportions,  then 
the  expansion  may  be  stated  in  words.  The  scalar  product 
of  two  vector  products  is  equal  to  the  (scalar)  product  of  the 
antecedents  times  the  (scalar)  product  of  the  consequents 
diminished  by  the  (scalar)  product  of  the  means  times  the 
(scalar)  product  of  the  extremes. 

To  reduce  a  vector  product  of  two  vectors  each  of  which 
is  itself  a  vector  product  of  two  vectors,  as 

(AxB)x(CxD). 
Let  CxD  =  E.     The  product  becomes 

(AxB)xE  =  A.E  B  -  B.E  A. 

Substituting  the  value  of  E  back  into  the  equation : 

(AxB)x(CxD)  =  (A.CxD)B  -  (B-CxD)  A.         (26) 
Let  F  =  AxB.     The  product  then  becomes 

Fx(CxD)  =  F»D  C-F.C  D 
(AxB)x(CxD)  =  (AxB»D)C  -  (AxB»C)  D.       (26)' 

By  equating  these  two  equivalent  results  and  transposing 
all  the  terms  to  one  side  of  the  equation, 

[B  C  D]  A  -  [C  D  A]  B  +  [D  A  B]  C  -  [A  B  C]  D  =  0.    (27) 

This  is  an  equation  with  scalar  coefficients  between  the  four 
vectors  A,  B,  C,  D.     There  is  in  general  only  one  such  equa- 


DIRECT  AND  SKEW  PRODUCTS  OF  VECTORS        77 

tion,  because  any  one  of  the  vectors  can  be  expressed  in  only 
one  way  in  terms  of  the  other  three :  thus  the  scalar  coeffi- 
cients of  that  equation  which  exists  between  four  vectors  are 
found  to  be  nothing  but  the  four  scalar  triple  products  of 
those  vectors  taken  three  at  a  time.  The  equation  may  also 
be  written  in  the  form 

[A  B  C  D]  =  [B  C  D]  A  +  [C  A  D]  B  +  [A  B  D]  C.     (27)' 

More  examples  of  reduction  formulae,  of  which  some  are 
important,  are  given  among  the  exercises  at  the  end  of  the 
chapter.  In  view  of  these  it  becomes  fairly  obvious  that 
the  combination  of  any  number  of  vectors  connected  in 
any  legitimate  way  by  dots  and  crosses  or  the  product  of  any 
number  of  such  combinations  can  be  ultimately  reduced  to 
a  sum  of  terms  each  of  which  contains  only  one  cross  at  most. 
The  proof  of  this  theorem  depends  solely  upon  analyzing  the 
possible  combinations  of  vectors  and  showing  that  they  all 
fall  under  the  reduction  formulae  in  such  a  way  that  the 
crosses  may  be  removed  two  at  a  time  until  not  more  than 
one  remains. 

*  42.]  The  formulae  developed  in  the  foregoing  article  have 
interesting  geometric  interpretations.  They  also  afford  a 
simple  means  of  deducing  the  formulae  of  Spherical  Trigo- 
nometry. These  do  not  occur  in  the  vector  analysis  proper. 
Their  place  is  taken  by  the  two  quadruple  products, 

(AxB).(CxD)  =  A.C  B-D  -  B.C  A.D         (25) 
and  (AxB)x(CxD)  =  [ACD]  B  -  [BCD]  A 

=  [ABD]  C  -  [ABC]  D,     (26) 

which  are  now  to  be  interpreted. 

Let  a  unit  sphere  (Fig.  22)  be  given.  Let  the  vectors 
A,  B,  C,  D  be  unit  vectors  drawn  from  a  common  origin,  the 
centre  of  the  sphere,  and  terminating  in  the  surface  of  the 
sphere  at  the  points  A,  B,  C,  D.  The  great  circular  arcs 


78 


VECTOR  ANALYSIS 


FIG.  22. 


AB,  A  (7,  etc.,  give  the  angles  between  the  vectors  A  and  B, 
A  and  C,  etc.  The  points  A,  B,  (7,  D  determine  a  quadrilateral 
upon  the  sphere.  A  C  and  BD  are  one 
pair  of  opposite  sides ;  A  D  and  B  C,  the 
other.  A  B  and  CD  are  the  diagonals. 

(AxB).(CxD)  =  A.C  B.D  -  A.D  B.C 

AxB  =  sin  (A,B),  CxD  =  sin(C,  D). 

The  angle  between  AxB  and  CxD  is  the 
angle  between  the  normals  to  the  AB- 
and  CD-planes.  This  is  the  same  as 

the  angle  between  the  planes  themselves.     Let  it  be  denoted 

by  x.     Then 

(AxB). (CxD)  =  sin  (A,B)  sin  (C,D)  cos  a;. 

The   angles   (A,  B),  (C,  D)   may  be   replaced  by  the   great 
circular  arcs  AB,  CD  which  measure  them.     Then 

(AxB).(CxD)  —  sin  A  B  sin  CD  cos  x, 
A.C  B.D  — A.D  B«C  =  cos  AC  cosBD  —  cos  AD  cos  BC. 

Hence 

sin  A  B  sin  CD  cos  x  =  cos  A  C  cos  B  D  —  cos  A  D  cos  B  C. 

In  words :  The  product  of  the  cosines  of  two  opposite  sides 
of  a  spherical  quadrilateral  less  the  product  of  the  cosines  of 
the  other  two  opposite  sides  is  equal  to  the  product  of  the 
sines  of  the  diagonals  multiplied  by  the 
cosine  of  the  angle  between  them.  This 
theorem  is  credited  to  Gauss. 

Let  A,  B,  C  (Fig.  23)  be  a  spherical  tri- 
angle, the  sides  of  which  are  arcs  of  great 
circles.  Let  the  sides  be  denoted  by  a,  &,  c 
respectively.  Let  A,  B,  C  be  the  unit  vectors 
drawn  from  the  center  of  the  sphere  to  the  points  A,  B,  C. 
Furthermore  let  p^  pb,  pc  be  the  great  circular  arcs  dropped 


FIG.  23. 


DIRECT  AND  SKEW  PRODUCTS   OF   VECTORS        79 

perpendicularly  from  the  vertices  A,  B,  C  to  the  sides  <z,  J,  c. 
Interpret  the  formula 

(AxB).(CxA)  =  A-C  B-A  -  B-C  A- A. 

(AxB)  =  sin  (A,  B)  =  sin  c,     (Cx A)  =  sin  (C,  A)  =  sin  b. 
Then  (AxB).  (CxA)  —  sin  c  sin  b  cos  x, 

where  x  is  the  angle  between  AxB  and  CxA.  This 
angle  is  equal  to  the  angle  between  the  plane  of  A,  B  and  the 
plane  of  C,  A.  It  is,  however,  not  the  interior  angle  A  which 
is  one  of  the  angles  of  the  triangle  :  but  it  is  the  exterior 
angle  180°  —  A,  as  an  examination  of  the  figure  will  show. 

Hence 

(AxB)x(CxA)  =  sin  c  sin  b  cos  (180°  —  A) 

=  —  sin  c  sin  b  cos  A 
A«C  B«A  —  B«C  A«  A  =  cos  b  cos  c  —  cos  a  1. 

By  equating  the  results  and  transposing, 

cos  a  =  cos  b  cos  c  —  sin  b  sin  c  cos  A 
cos  b  ='  cos  c  cos  a  —  sin  c  sin  a  cos  B 
cos  c  =  cos  a  cos  b  —  sin  a  sin  b  cos  C. 

The  last  two  may  be  obtained  by  cyclic  permutation  of  the 
letters  or  from  the  identities 

(BxC)x(AxB)  =  B.A  C.B  -  C.A, 

(CxA)x(BxC)  =  C-B  A.C  -  B.C. 

Next  interpret  the  identity  (AxB)x(CxD)  in  the  special 
cases  in  which  one  of  the  vectors  is  repeated. 

(AxB)x(AxC)  =  [A  B  C]  A. 

Let  the  three  vectors  a,  b,  c  be  unit  vectors  in  the  direction  of 
BxC,  CxA,  AxB  respectively.  Then 

AxB  =  c  sin  c,  AxC  =  —  b  sin  b 

(AxB)x(AxC)  =  —  cxb  sin  c  sin  b  —  A  sin  c  sin  b  sin  A 
[A  B  C]  =  (AxB).C  =  c.C  sin  c  =  cos  (90°  —pe)  sin  c 
[ABC]  A  =  sin  c  sin pe • 


80  VECTOR  ANALYSIS 

By  equating  the  results  and  cancelling  the  common  factor, 

sin  pe  =  sin  b  sin  A 
=  sin  c  sin  B 
=  sin  a  sin  C. 


The  last  two  may  be  obtained  by  cyclic  permutation  of  the 
letters.  The  formulae  give  the  sines  of  the  altitudes  of  the 
triangle  in  terms  of  the  sines  of  the  angle  and  sides.  Again 

write 

(AxB)x(AxC)  =  [ABC]A 

(BxC)x(BxA)  =  [BCA]B 
(CxA)x(CxB)  =  [C  AB]  C. 

Hence  sin  c  sin  b  sin  A-—  [A  B  C] 

sin  a  sin  c  sin  B  =  [B  C  A] 
sin  b  sin  a  sin  0  =  [C  A  B]. 

The  expressions  [ABC],  [BCA],  [CAB]  are  equal.  Equate 
the  results  in  pairs  and  the  formulae 

sin  b  sin  A  =  sin  a  sin  B 
sin  c  sin  B  =  sin  b  sin  C 
sin  a  sin  C  •*=  sin  c  sin  A 

are  obtained.    These  may  be  written  in  a  single  line. 

sin  A  _  sin  B  _  sin  C 
sin  a       sin  b       sin  c 

The  formulae  of  Plane  Trigonometry  are  even  more  easy  to 
obtain.  If  A  B  C  be  a  triangle,  the  sum  of  the  sides  taken 
as  vectors  is  zero  —  for  the  triangle  is  a  closed  polygon. 
From  this  equation 

a  +  b  +  c  =  0 

almost  all  the  elementary  formulae  follow  immediately.  It 
is  to  be  hoticed  that  the  angles  from  a  to  b,  from  b  to  c,  from 


DIRECT  AND  SKEW  PRODUCTS   OF   VECTORS        81 

c  to  a  are  not  the  interior  angles  A,  B,  C,  but  the  exterior 
angles  ISO0 -A,  180°  -  B,  180°  -  C. 

—  a  =  b  +  c 
a.a  =  (b  +  c).(b  +  c)  =  b»b  +  e«c  +  2  b«c. 

If  a,  &,  c  be  the  length  of  the  sides  a,  b,  c.  this  becomes 

a2  =  J2  -f  c2  —  2  b  c  cos  A 
b2  =  c2  -f  a2  —  2  c  a  cos  5 
c2  =  a2  +  62  —  2  a  b  cos  (7. 

The  last  two  are  obtained  in  a  manner  similar  to  the  first 
one  or  by  cyclic  permutation  of  the  letters. 
The  area  of  the  triangle  is 

1         b  —  -b         —  i    - 

2  a  b  sin  C  =  %  b  c  sin  A  =  ^  c  a  sin  B. 

If  each  of  the  last  three  equalities  be  divided  by  the  product 
|  a  b  c,  the  fundamental  relation 

sin  A      sin  B      sin  C 
a  b  c 

is  obtained.    Another  formula  for  the  area  may  be  found  from 

the  product 

(bxc)-(bxc)  =  (cxa)«(axb) 

2  Area  (b  c  sin  A)  =  (c  a  sin  B}  (a  b  sin  G) 

a2  sin  B  sin  G 
2  Area  =  — 


Reciprocal  Systems  of  Three  Vectors.     Solution  of  Equations 

43.]  The  problem1  of  expressing  any  vector  r  in  terms  of 
three  non-coplanar  vectors  a,  b,  c  may  be  solved  as  follows. 
Let 


82  VECTOR  ANALYSIS 

where  a,  6,  c  are  three  scalar   constants  to  be  determined. 

Multiply  by  •  b  x  c. 

//• 
r-bxc  =  a  a«bxc  +  &  b-bxc  +  cc-bxc 

or  [rbc]=a[abo]. 

In  like  manner  by  multiplying  the  equation  by  -  c  X  a  and 
.  a  x  b  the  coefficients  &  and  c  may  be  found. 

[r  c  a]  =  &  [b  c  a] 
[r  a  b]  =  c  [c  a  b] 


c]         [r  c  a]      ,[r  a  b] 
Hence  r  =  ±-—4.  a  +  ^-—\  b  +  -^—  r^  c.         (28) 

[a  be]         [be  aj         [c  a  b  J 

The    denominators   are   all    equal.      Hence   this   gives   the 
equation 

[a  b  c]  r  -  [b  c  r]  a  +  [c  r  a]  b  -  [r  a  b]  c  =  0 

which  must  exist  between  the  four  vectors  r,  a,  b,  c. 
The  equation  may  also  be  written 

r«b  x  c        r«cxa        r«axb 
[abc]  [abc]  [a  be] 

bxc  cxa,  axb 

or  r  =  r  •   --  a  +  r  •  —   —  b  +  r  •  —    —  c. 

[abc]  [abc]  [abc] 

The  three  vectors  which  appear  here  multiplied  by  r«,  namely 

bxc        cxa        axb 

_  »      _  j     _ 

[a  be]       [a  b  c]       [a  b  c] 

are  very  important.     They  are  perpendicular  respectively  to 
the  planes  of  b  and  c,  c  and  a,  a  and  b.     They  occur  over  and 
over  again  in  a  large  number  of  important  relations.     For 
this  reason  they  merit  a  distinctive  name  and  notation. 
Definition  :  The  system  of  three  vectors 

b  x  c      cxa      axb 
[abc]'    [abc]'    [abc] 


DIRECT  AND  SKEW  PRODUCTS  OF  VECTORS        83 

which  are  found  by  dividing  the  three  vector  products  b  x  c, 
c  x  a,  a  x  b  of  three  non-coplanar  vectors  a,  b,  c  by  the  scalar 
product  [  a  b  c  ]  is  called  the  reciprocal  system  to  a,  b,  c. 

The  word  non-coplanar  is  important.  If  a,  b,  c  were  co- 
planar  the  scalar  triple  product  [a  b  c]  would  vanish  and 
consequently  the  fractions 

bxc       cxa       axb 

>    »    

[a be]       [a b  c]       [a b  c] 

would  all  become  meaningless.  Three  coplanar  vectors  have 
no  reciprocal  system.  This  must  be  carefully  remembered. 
Hereafter  when  the  term  reciprocal  system  is  used,  it  will  be 
understood  that  the  three  vectors  a,  b,  c  are  not  coplanar. 
The  system  of  three  vectors  reciprocal  to  system  a,  b,  c 
will  be  denoted  by  primes  as  a',  b',  c'. 

bxc  cxa  axb  (29) 

a— —    —  '    b  = —    — '    c  = 

[a  be]  [a  b  c]  [a  b  c] 

The  expression  for  r  reduces  then  to  the  very  simple  form 
r  =  r.a'a  +  r.b'b  +  r.c'c.  (30) 

Tl^e  vector  r  may  be  expressed  in  terms  of  the  reciprocal 
system  a',  b',  c'  instead  of  in  terms  of  a,  b,  c.  In  the  first 
place  it  is  necessary  to  note  that  if  a,  b,  c  are  non-coplanar, 
a',  b',  c'  which  are  the  normals  to  the  planes  of  b  and  c, 
c  and  a,  a  and  b  must  also  be  non-coplanar.  Hence  r  may 
be  expressed  in  terms  of  them  by  means  of  proper  scalar 
coefficients  #,  y,  z. 

r  =  x  a'  +  7/b'  +  z  c' 

or  [abc]r  =  zbxc  +  2/cxa  +  zaxb. 

Multiply  successively  by  ^a,  »b,  «c.     This  gives 

[a  b  c]  r«a  =  x  [b  c  a],       x  —  r»a 
[a be]  r«b  —  y  [c  ab],       y  =  r«b 
[a  b  c]  r»c  =  z  [a  b  c],       z  =  r»c 
Hence  r  =  r«aa'  +  r«bb'  +  r«c  c'.  (31) 


84  VECTOR  ANALYSIS 

44.]  If  a',  b',  c'  be  the  system  reciprocal  to  a,  b,  c  the 
scalar  product  of  any  vector  of  the  reciprocal  system  into  the 
corresponding  vector  of  the  given  system  is  unity;  but 
the  product  of  two  non-corresponding  vectors  is  zero.  That  is 

a'.a  =  b'-b  =  c'.c  =  l  (32) 

a'.b  =  a'.c  =  b'.a  =  b'-c  =  c'»a  =  c'-b  =  0. 

This  may  be  seen  most  easily  by  expressing  a',  b',  c'  in 
terms  of  themselves  according  to  the  formula  (31) 

r  =  r»a  a'  +  r«b  b'  +  r«c  c'. 
Hence  a'  =  a'«aa'  +  a'-bb'  +  a'»"c  c' 

b'  =  b'.aa'  +  b'-bb'  +  b'-cc' 
o'  =  c'«aa'  +  c'-bb'  +  c'.cc'. 

Since  a',  b',  c'  are  non-coplanar  the  corresponding  coeffi- 
cients on  the  two  sides  of  each  of  these  three  equations  must 
be  equal.  Hence  from  the  first 

1  =  a'.a  0  =  a'.b  0  =  a'  c. 
From  the  second  0  =  b'«a  l=b'-b  0  —  b'.c. 
From  the  third  O^c'.a  0  =  c'.b  l  =  c'«c. 

This  proves  the  relations.  They  may  also  be  proved 
directly  from  the  definitions  of  a',  b',  c'. 

,  bxc  b  x  c  •  a      [be  a] 

a'»a  = •  a  = =- -  =  1 

[abc]  [abc  ]       [a be] 

bxc  bxc.b          0 

a '  •  b  = .  b  =  = =0 

[abc]  [abc]         [abc] 

and  so  forth. 

Conversely  if  two  sets  of  three  vectors  each,  say  A,  B,  C, 
and  a,  b,  c,  satisfy  the  relations 

A«a  =  B'b  =  C»c  =  1 
A.a  =  A.c  =  B«a  =  B.C  =  C-a  =  (M>  =  0 


DIRECT  AND  SKEW  PRODUCTS   OF  VECTORS        85 

then  the  set  A,  B,  C  is  the  system  reciprocal  to  a,  b,  c. 
By  reasoning  similar  to  that  before 

A  =  A-a  a'  +  A«b  b'  +  A«c  c' 


C  =  C«a  a'  +  C«b  b'  +  C-c  c'. 

Substituting  in  these   equations  the  given  relations  the  re- 

sult is 

A  =  a',     B  =  b',     C  =  c'. 
Hence 

Theorem  :  The  necessary  and  sufficient  conditions  that  the 
set  of  vectors  a',  b',  c'  be  the  reciprocals  of  a,  b,  c  is  that 
they  satisfy  the  equations  4 

a'.a  =  b'.b  =  c'-c  =  l  (32) 

a'.b  =  a'.c  =  b'.a  =  b'.c  =  c'.a  =  c'.b  =  0. 

As  these  equations  are  perfectly  symmetrical  with  respect 
to  a',  b',  c'  and  a,  b,  c  it  is  evident  that  the  system  a,  b,  c  may 
be  looked  upon  as  the  reciprocal  of  the  system  a',  b',  c'  just 
as  the  system  a',  b',  c'  may  be  regarded  as  the  reciprocal  of 
a,  b,  c.  That  is  to  say, 

Theorem:  If  a',  b',  c'  be  the  reciprocal  system  of  a,  b,  c, 
then  a,  b,  c  will  be  the  reciprocal  system  of  a',  b',  c'. 

b'  x  c'  t  c'  x  a'  ^      _  a'  x  b'       (29)' 

"  [a'bV]'       "[a'b'c']'       ~[a'b'c']' 

These  relations  may  be  demonstrated  directly  from  the. 
definitions  of  a',  b',  c'.  The  demonstration  is  straightfor- 
ward, but  rather  long  and  tedious  as  it  depends  on  compli- 
cated reduction  formulae.  The  proof  given  above  is  as  short 
as  could  be  desired.  The  relations  between  a',  b',  c'  and 
a,  b,  c  are  symmetrical  and  hence  if  a',  b',  c'  is  the  reciprocal 
system  of  a,  b,  c,  then  a,  b,  c  must  be  the  reciprocal  system  of 
a',  b',  c'. 


86  VECTOR  ANALYSIS 

45.]  Theorem :  If  a ' ,  b ' ,  c '  and  a,  b,  c  be  reciprocal  systems 
the  scalar  triple  products  [a'bV]  and  [a  b  c]  are  numerical 
reciprocals.  That  is 

[a'bV]   [abc]=l  (33; 

,    ,    ,,      f  b  x  c     ex  a      axbl 
~L[abc]     [abc]     [abc]J 

=  [bxc  cxa  axb]. 

[abc]3 

[bxc  cxa  axb]  =  (bxc)x(cxa>(axb). 

(bxc)  x  (cxa)  =  [abc]c. 
[bxc  cxa  axb]  =  [abc]  c«axb  =  [abc]2. 
1  1 


But 
Hence 

Hence 


[a'bV]  = 


[abc]2  = 


(33y 


[abc]3  [abc] 

By  means  of  this  relation  between  [a'  b'  c']  and  [a  b  c]  it 
is  possible  to  prove  an  important  reduction  formula, 

P.A    P.B    P-C 


(P.QxE)(A-BxC)  = 


Q.A    Q.B 

E.A  E.B 


Q.C 
E-C 


(34) 


which  replaces  the  two  scalar  triple  products  by  a  sum  of 
nine  terms  each  of  which  is  the  product  of  three  direct  pro- 
ducts. Thus  the  two  crosses  which  occur  in  the  two  scalar 
products  are  removed.  To  give  the  proof  let  P,  Q,  E  be 
expressed  as 

p  =  P.A  A'  +  P.B  B'  +  p.c  cr 

E  =  E.A  A'  +  E.B  B'  +  E.C  c'. 
P.A   P.B    p.c 

Then  FPQE1=    Q-A     Q.B     Q-C 

E.A  E.B 


But 


[A'B'C'J  = 


E.C 
1 

[ABC] 


[A'B'C']. 


DIRECT  AND  SKEW  PRODUCTS   OF  VECTORS        87 


Hence          [PftR]  [ABC]  = 


P.A     P.B      P.C 

Q.A   Q.B    Q.C 

S.A     B-B     R.C 


The  system  of  three  unit  vectors  i,  j,  k  is  its  own  reciprocal 
system. 

.,_jxk  _i_.    ,,_kxi          k,_ixj_ 
---  --J'         - 


For  this  reason  the  primes  i',  j',  k'  are  not  needed  to  denote 
a  system  of  vectors  reciprocal  to  i,  j,  k.  The  primes  will 
therefore  be  used  in  the  future  to  denote  another  set  of  rect- 
angular axes  i,  j,  k  ,  just  as  X]  ',  F',  Z1  are  used  to  denote  a 
set  of  axes  different  from  JT,  Z,  Z. 

TJie  only  systems  of  three  vectors  which  are  their  own  reciprocals 
are  the  right-handed  and  left-handed  systems  of  three  unit 
vectors.  That  is  the  system  i,  j,  k  and  the  system  i,  j,  —  k. 

Let  A,  B,  C  be  a  set  of  vectors  which  is  its  own  reciprocal. 
Then  by  (32) 

A.A  =  B.B  =  c-c  =  i. 

Hence  the  vectors  are  all  unit  vectors. 

A.B  =  A.C  =  o. 

Hence  A  is  perpendicular  to  B  and  C. 

B.A  =  B-C  =  o. 

Hence  B  is  perpendicular  to  A  and  C. 

C.A  =C-B  =  O. 

Hence  C  is  perpendicular  to  A  and  B. 

Hence  A,  B,  C  must  be  a  system  like  i,  j,  k  or  like  i,j,  —  k. 

*  46.]  A  scalar  equation  of  the  first  degree  in  a  vector  r  is 
an  equation  in  each  term  of  which  r  occurs  not  more  than 
once.  The  value  of  each  term  must  be  scalar.  As  an  exam- 
ple of  such  an  equation  the  following  may  be  given. 

a  a-bxr  +  &(cxd).(exr)  +  c  f«r  -f  d  =  0, 


88  VECTOR  ANALYSIS 

where  a,  b,  c,  d,  e,  f  are  known  vectors ;  and  a,  Z>,  c,  d,  known 
scalars.  Obviously  any  scalar  equation  of  the  first  degree  in 
an  unknown  vector  r  may  be  reduced  to  the  form 

r-A  =  a 

where  A  is  a  known  vector ;  and  a,  a  known  scalar.  To  ac- 
complish this  result  in  the  case  of  the  given  equation  proceed 

as  follows. 

a  axb»r  +  b  (cxd)xe-r  +  c  f»r  +  d  —  0 

{a  axb  +  b  (cxd)xe  +  c  f}«r  =  —  d. 

In  more  complicated  forms  it  may  be  nec'essary  to  make  use 
of  various  reduction  formulae  before  the  equation  can  be  made 

to  take  the  desired  form, 

r»A  =  a. 

As  a  vector  has  three  degrees  of  freedom  it  is  clear  that  one 
scalar  equation  is  insufficient  to  determine  a  vector.  Three 
scalar  equations  are  necessary. 

The  geometric  interpretation  of  the  equa- 
tion 

r-A  =  a  (36) 

'A 

is  interesting.     Let  r  be  a  variable  vector 
(Fig.  24)  drawn  from  a  fixed  origin.     Let 
A  be  a  fixed  vector  drawn  from  the  same 
origin.     The  equation  then  becomes 

r  A  cos  (r,  A)  =  a, 

or  r  cos  (r,A)  =—. , 

A 

if  r  be  the  magnitude  of  r ;  and  A  that  of  A.     The  expression 

r  cos  (r,  A) 

is  the  projection  of  r  upon  A.  The  equation  therefore  states 
that  the  projection  of  r  upon  a  certain  fixed  vector  A  must 


DIRECT  AND  SKEW  PRODUCTS   OF  VECTORS       89 

always  be  constant  and  equal  to  a!  A.  Consequently  the  ter- 
minus of  r  must  trace  out  a  plane  perpendicular  to  the  vector 
A  at  a  distance  equal  to  a/A  from  the  origin.  The  projec- 
tion upon  A  of  any  radius  vector  drawn  from  the  origin  to  a 
point  of  this  plane  is  constant  and  equal  to  a/A.  This  gives 
the  following  theorem. 

Theorem  :  A  scalar  equation  in  an  unknown  vector  may  be 
regarded  as  the  equation  of  a  plane,  which  is  the  locus  of  the 
terminus  of  the  unknown  vector  if  its  origin  be  fixed. 

It  is  easy  to  see  why  three  scalar  equations  in  an  unknown 
vector  determine  the  vector  completely.  Each  equation  de- 
termines a  plane  in  which  the  terminus  of  r  must  lie.  The 
three  planes  intersect  in  one  common  point.  Hence  one  vec- 
tor r  is  determined.  The  analytic  solution  of  three  scalar 
equations  is  extremely  easy.  If  the  equations  are 

r«A  —  a 

r-B  =  I  (37) 

r»C  =  c, 

it  is  only  necessary  to  call  to  mind  the  formula 


Hence  r  =  a  A'  +  6  B'  +  c  C'.  (38) 

The  solution  is  therefore  accomplished.  It  is  expressed  in 
terms  A',  B',  C'  which  is  the  reciprocal  system  to  A,  B,  C.  One 
caution  must  however  be  observed.  The  vectors  A,  B,  C  will 
have  no  reciprocal  system  if  they  are  coplanar.  Hence  the 
solution  will  fail.  In  this  case,  however,  the  three  planes  de- 
termined by  the  three  equations  will  be  parallel  to  a  line. 
They  will  therefore  either  not  intersect  (as  in  the  case  of  the 
lateral  faces  of  a  triangular  prism)  or  they  will  intersect  in  a 
common  line.  Hence  there  will  be  either  no  solution  for  r  or 
there  will  be  an  infinite  number. 


90  VECTOR  ANALYSIS 

From  four  scalar  equations 

r«A  =  a 

r.B  =  b  (39) 

r»C  =  c 

r.D=d 

the  vector  r  may  be  entirely  eliminated.  To  accomplish  this 
solve  three  of  the  equations  and  substitute  the  value  in  the 
fourth. 


or        a  [BCD]  +  &  [CAD]  +c  [ABD]  =  d  [ABC].     (40) 

*  47.]  A  vector  equation  of  the  first  degree  in  an  unknown 
vector  is  an  equation  each  term  of  which  is  a  vector  quantity 
containing  the  unknown  vector  not  more  than  once.  Such 
an  equation  is 

(AxB)x(Cxr)  +  D  ET  +  n  r  +  F  =0, 

where  A,  B,  C,  D,  E,  F  are  known  vectors,  n  a  known  scalar, 
and  r  the  unknown  vector.  One  such  equation  may  in  gen- 
eral be  solved  for  r.  That  is  to  say,  one  vector  equation  is  in 
general  sufficient  to  determine  the  unknown  vector  which  is 
contained  in  it  to  the  first  degree. 

The  method  of  solving  a  vector  equation  is  to  multiply  it 
with  a  dot  successively  by  three  arbitrary  known  non-coplanar 
vectors.  Thus  three  scalar  equations  are  obtained.  These 
may  be  solved  by  the  methods  of  the  foregoing  article.  In  the 
first  place  let  the  equation  be 

A  a*r  +  B  b-r  +  C  c»r  =  D, 

where  A,  B,  C,  D,  a,  b,  c  are  known  vectors.  No  scalar  coeffi- 
cients are  written  in  the  terms,  for  they  may  be  incorporated  in 
the  vectors.  Multiply  the  equation  successively  by  A',  B',  C'. 
It  is  understood  of  course  that  A,  B,  C  are  non-coplanar. 


DIRECT  AND  SKEW  PRODUCTS   OF  VECTORS        91 

a.r  =  D-Ar 
b«r  =  D-B' 


But  r  =  a'  a«r  +  b'  b«r  +  c'  c-r. 

Hence  r  =  D-A'  a'  +  D-B'  b'  +  D-C'  c'. 

The  solution  is  therefore  accomplished  in  case  A,  B,  C  are  non- 
coplanar  and  a,  b,  c  also  non-coplanar.  The  special  cases  in 
which  either  of  these  sets  of  three  vectors  is  coplanar  will  not 
be  discussed  here. 

The  most  general  vector  equation  of  the  first  degree  in  an 
unknown  vector  r  contains  terms  of  the  types 

A  a«r.     n  r,     Exr,     D. 

That  is  it  will  contain  terms  which  consist  of  a  known 
vector  multiplied  by  the  scalar  product  of  another  known  vec- 
tor and  the  unknown  vector  ;  terms  which  are  scalar  multi- 
ples of  the  unknown  vector;  terms  which  are  the  vector 
product  of  a  known  and  the  unknown  vector  ;  and  constant 
terms.  The  terms  of  the  type  A  a«r  may  always  be  reduced 
to  three  in  number.  For  the  vectors  a,  b,  c,  •  •  •  which  are 
multiplied  into  r  may  all  be  expressed  in  terms  of  three  non- 
coplanar  vectors.  Hence  all  the  products  a»r,  b-r,  c«r,  •  •  • 
may  be  expressed  in  terms  of  three.  The  sum  of  all  terms  of 
the  type  A  a«r  therefore  reduces  to  an  expression  of  three 

terms,  as 

A  a»r  +  B  b-r  +  C  c«r. 

The  terms  of  the  types  n  r  and  Exr  may  also  be  expressed 
in  this  form. 

n  r  =  wa'a«r  +  n  b'b«r  +  n  c'c«r 
Exr  ==  Exa'  a«r  +  Exb'  b-r+Exc'  c«r. 

Adding  all  these  terms  together  the  whole  equation  reduces 

to  the  form 

L  a»r  +  M  b»r  +  N  c«r  =  K. 


92  VECTOR  ANALYSIS 

This  has  already  been  solved  as 

r  =  K.L'  a'  +  K.M'  b'  +  K-N'  c'. 

The  solution  is  in  terms  of  three  non-coplanar  vectors  a',  b',  c'. 
These  form  the  system  reciprocal  to  a,  b,  c  in  terms  of  which 
the  products  containing  the  unknown  vector  r  were  expressed. 

*  SUNDRY  APPLICATIONS  OF  PRODUCTS 

Applications  to  Mechanics 

48.]  In  the  mechanics  of  a  rigid  body  a  force  is  not  a 
vector  in  the  sense  understood  in  this  book.  See  Art.  3. 
A  force  has  magnitude  and  direction ;  but  it  has  also  a  line 
of  application.  Two  forces  which  are  alike  in  magnitude 
and  direction,  but  which  lie  upon  different  lines  in  the  body 
do  not  produce  the  same  effect.  Nevertheless  vectors  are 
sufficiently  like  forces  to  be  useful  in  treating  them. 

If  a  number  of  forces  fv  f2,  fg,  •••act  on  a  body  at  the 
.same  point  0,  the  sum  of  the  forces  added  as  vectors  is  called 
the  resultant  R. 

R  =  f1  +  fa  +  f,  +  ... 

In  the  same  way  if  f  x,  f  2,  f  3  •  •  •  do  not  act  at  the  same  point 
the  term  resultant  is  still  applied  to  the  sum  of  these  forces 
added  just  as  if  they  were  vectors. 

R  =  f1  +  fa +  £,  +  ...  (41) 

The  idea  of  the  resultant  therefore  does  not  introduce  the 
line  of  action  of  a  force.  As  far  as  the  resultant  is  concerned 
a  force  does  not  differ  from  a  vector. 

Definition :  The  moment  of  a  force  f  about  the  point  0  is 
equal  to  the  product  of  the  force  by  the  perpendicular  dis- 
tance from  0  to  the  line  of  action  of  the  force.  The  moment 
however  is  best  looked  upon  as  a  vector  quantity.  Its  mag- 
nitude is  as  defined  above.  Its  direction  is  usually  taken  to 


DIRECT  AND  SKEW  PRODUCTS   OF  VECTORS       93 

be  the  normal  on  that  side  of  the  plane  passed  through  the 
point  0  and  the  line  f  upon  which  the  force  appears  to  pro- 
duce a  tendency  to  rotation  about  the  point  0  in  the  positive 
trigonometric  direction.  Another  method  of  defining  the 
moment  of  a  force  f  =  P  Q  about  the  point  0  is  as  follows  : 
The  moment  of  the  force  f  =  P  Q  about  the  point  0  is  equal 
to  twice  the  area  of  the  triangle  0  P  Q.  This  includes  at  once 
both  the  magnitude  and  direction  of  the  moment  (Art.  25). 
The  point  P  is  supposed  to  be  the  origin  ;  and  the  point  Q, 
the  terminus  of  the  arrow  which  represents  the  force  f.  The 
letter  M  will  be  used  to  denote  the  moment.  A  subscript  will 
be  attached  to  designate  the  point  about  which  the  moment  is 
taken. 

Mo  {t}  = 


The  moment  of  a  number  of  forces  f  x,  f2,  •  •  •  is  the  (vector) 
sum  of  the  moments  of  the  individual  forces. 


If  *i  =  -Pi0i.      **  =  ?*Qi'~ 

Mo  {f n  fa,  •  •  •  }  -  2  (0  P!  Ql  +  OPZ  Qz  +  • ..). 

This  is  known  as  the  total  or  resultant  moment  of  the  forces 

f  1»  *2'  "  '  ' 

49.]  If  f  be  a  force  acting  on  a  body  and  if  d  be  the  vector 
drawn  from  the  point  0  to  any  point  in  the  line  of  action  of 
the  force,  the  moment  of  the  force  about  the  point  0  is  the 
vector  product  of  d  into  f . 

Mo  W  -  dxf  (42) 

For  dxf  =  d/sin  (d,f)  e, 

if  e  be  a  unit  vector  in  the  direction  of  dxf. 
dxf  =  d  sin  (d,  f)/e. 

Now  d  sin  (d,  f)  is  the  perpendicular  distance  from  0  to  f. 
The  magnitude  of  dxf  is  accordingly  equal  to  this  perpen- 
dicular distance  multiplied  by  /,  the  magnitude  of  the  force. 


94  VECTOR  ANALYSIS 

This  is  the  magnitude  of  the  moment  MO  {f}.     The  direction 
of  dxf  is  the  same  as  the  direction  of  the  moment.     Hence 

the  relation  is  proved. 

Mo  {f}  =  dxf. 

The  sum  of  the  moments  about  0  of  a  number  of  forces 
f  r  f  2,  •  •  •  acting  at  the  same  point  P  is  equal  to  the  moment 
of  the  resultant  E  of  the  forces  acting  at  that  point.  For  let 
d  be  the  vector  from  0  to  P.  Then 

Mo  {f  !>  =  dxf  l 
Mo  {fa|  -  dxf2  . 


Mo  {fJ  +  Ko  {fa}  +  ---  =  dxf1  +  dxfa  +  ...  (43) 

f    H  ----  ) 


The  total  moment  about  0'  of  any  number  of  forces  f  x,  f  2,  •  •  • 
acting  on  a  rigid  body  is  equal  to  the  total  moment  of  those 
forces  about  0  increased  by  the  moment  about  0'  of  the 
resultant  Ro  considered  as  acting  at  0. 

Mo-  {f  !,  f  a  ,•••}  =  Mo  {f  p  f  jp  •  •  •}  +  M0'  (Ho  \.      (44) 

Let  d15  d2,  •  •  •  be  vectors  drawn  from  0  to  any  point  in 
f  j,  f2,  •  •  •  respectively.  Let  d/,  d2',  •  •  •  be  the  vectors  drawn 
from  0'  to  the  same  points  in  f  v  f2,  •  •  •  respectively.  Let  c 
be  the  vector  from  0  to  0'.  Then 

d1  =  d1'  +  c,          d2  =  d2'  +  c,  ••• 

Mo  {f  !,  f2,  •  •  •}  =  dxxf  j  +  d2xf2  +  .  -  . 

M0'  #!,  f2,  •  •  -\  =  d/xf  !  +  d2'xf2  +  •  •  . 

=  (dx  -  c)xf  !  +  (d2  -  c)xf  2  +  •  •  • 
=  djXf  i+daxf  2  +  ----  cx(f  j  +  f  2  +  •  •  •) 

But  —  c  is  the  vector  drawn  from  0'  to  0.  Hence  —  c  x  f, 
is  the  moment  about  0'  of  a  force  equal  in  magnitude  and 
parallel  hi  direction  to  f  1  but  situated  at  0.  Hence 


DIRECT  AND  SKEW  PRODUCTS   OF   VECTORS       95 

-cxCfj+fjj  +  ...)=-  cxR0  =  M0'  {Ro}- 
Hence  M0/  {f  1?  fa,  •  •  •}  =  M0  {f  15  f  2  ,•••}  +  M0'  {Ro  |. 


The  theorem  is  therefore  proved. 

The  resultant  R  is  of  course  the  same  at  all  points.  The 
subscript  0  is  attached  merely  to  show  at  what  point  it  is 
supposed  to  act  when  the  moment  about  0'  is  taken.  For 
the  point  of  application  of  R  affects  the  value  of  that  moment. 

The  scalar  product  of  the  total  moment  and  the  resultant 
is  the  same  no  matter  about  what  point  the  moment  be  taken. 
In  other  words  the  product  of  the  total  moment,  the  result- 
ant, and  the  cosine  of  the  angle  between  them  is  invariant 
for  all  points  of  space. 

B-M0'{fi,fa,  •••}=»•  Mo  {fi,*2»-"} 

where  0'  and  0  are  any  two  points  in  space.  This  important 
relation  follows  immediately  from  the  equation 

Me?  {fj,  fa,  •  •  •}  =  Mo  {fj  ,  f2,  .  •  •}  +  M0'  {Ro}. 
ForE-Mo^f^fa,  •••}=£.  Mo  {f^f-j,  •  •  •}  +  R  •  M0'  {R0}- 

But  the  moment  of  R  is  perpendicular  to  R  no  matter  what 
the  point  0  of  application  be.  Hence 

R.Mo'  IRo}  =  0 

and  the  relation  is  proved.  The  variation  in  the  total 
moment  due  to  a  variation  of  the  point  about  which  the 
moment  is  taken  is  always  perpendicular  to  the  resultant. 

50.]  A  point  0'  may  be  found  such  that  the  total  moment 
about  it  is  parallel  to  the  resultant.  The  condition  for 
parallelism  is 

£.•   =0 


RxM0'  |f  i  ,  fa,  •  •  •}  =  RxM0  {£lf  fa,  •  •  •} 
+  R  X  M0'  |B0  1  =  0 


96  VECTOR  ANALYSIS 

where  0  is  any  point  chosen  at  random.  Replace  M0'{EO} 
by  its  value  and  for  brevity  omit  to  write  the  f  v  f  2,  •  •  •  in  the 
braces  { }.  Then 

ExM(y  =  ExM0  —  Ex(cxE)  =  0. 
The  problem  is  to  solve  this  equation  for  c. 

ExM0  —  R'B.  c  +  E«c  E  =  0. 

Now  E  is  a  known  quantity.  MO  is  also  supposed  to  be 
known.  Let  c  be  chosen  in  the  plane  through  0  perpen- 
dicular to  E.  Then  E-c  =  0  and  the  equation  reduces  to 

ExM0  —  B«B  c 


c  = 


E-E 

If  c  be  chosen  equal  to  this  vector  the  total  moment  about 
the  point  0  ',  which  is  at  a  vector  distance  from  0  equal  to  c, 
will  be  parallel  to  E.  Moreover,  since  the  scalar  product  of 
the  total  moment  and  the  resultant  is  constant  and  since  the 
resultant  itself  is  constant  it  is  clear  that  in  the  case  where 
they  are  parallel  the  numerical  value  of  the  total  moment 
will  be  a  minimum. 

The  total  moment  is  unchanged  by  displacing  the  point 
about  which  it  is  taken  in  the  direction  of  the  resultant. 

For  M0'  |fi,fa,  •••}=M0{f1,fa,  •••}  -  cxB. 

If  c  =  0  (Jf  is  parallel  to  E,  cxE  vanishes  and  the  moment 
about  0'  is  equal  to  that  about  0.  Hence  it  is  possible  to 
find  not  merely  one  point  0'  about  which  the  total  moment 
is  parallel  to  the  resultant  ;  but  the  total  moment  about  any 
point  in  the  line  drawn  through  0  '  parallel  to  E  is  parallel 
to  E.  Furthermore  the  solution  found  in  equation  for  c  is 
the  only  one  which  exists  in  the  plane  perpendicular  to  E  — 
unless  the  resultant  E  vanishes.  The  results  that  have  been 
obtained  may  be  summed  up  as  follows  : 


DIRECT  AND  SKEW  PRODUCTS   OF  VECTORS        97 

If  any  system  of  forces  f  1?  f2,  •••  whose  resultant  is  not 
zero  act  upon  a  rigid  body,  then  there  exists  in  space  one 
and  only  one  line  such  that  the  total  moment  about  any 
point  of  it  is  parallel  to  the  resultant.  This  line  is  itself 
parallel  to  the  resultant.  The  total  moment  about  all  points 
of  it  is  the  same  and  is  numerically  less  than  that  about  any 
other  point  in  space. 

This  theorem  is  equivalent  to  the  one  which  states  that 
any  system  of  forces  acting  upon  a  rigid  body  is  equivalent 
to  a  single  force  (the  resultant)  acting  in  a  definite  line  and 
a  couple  of  which  the  plane  is  perpendicular  to  the  resultant 
and  of  which  the  moment  is  a  minimum.  A  system  of  forces 
may  be  reduced  to  a  single  force  (the  resultant)  acting  at  any 
desired  point  0  of  space  and  a  couple  the  moment  of  which 
(regarded  as  a  vector  quantity)  is  equal  to  the  total  moment 
about  0  of  the  forces  acting  on  the  body.  But  in  general  the 
plane  of  this  couple  will  not  be  perpendicular  to  the  result- 
ant, nor  will  its  moment  be  a  minimum. 

Those  who  would  pursue  the  study  of  systems  of  forces 
acting  on  a  rigid  body  further  and  more  thoroughly  may 
consult  the  Traite  de  Mechanique  Rationelle1  by  P.  APPELL. 
The  first  chapter  of  the  first  volume  is  entirely  devoted  to 
the  discussion  of  systems  of  forces.  Appell  defines  a  vector 
as  a  quantity  possessing  magnitude,  direction,  and  point  of 
application.  His  vectors  are  consequently  not  the  same  as 
those  used  in  this  book.  The  treatment  of  his  vectors  is 
carried  through  in  the  Cartesian  coordinates.  Each  step 
however  may  be  easily  converted  into  the  notation  of  vector 
analysis.  A  number  of  exercises  is  given  at  the  close  of 
the  chapter. 

51.]  Suppose  a  body  be  rotating  about  an  axis  with  a  con- 
stant angular  velocity  a.  The  points  in  the  body  describe 
circles  concentric  with  the  axes  in  planes  perpendicular  to 

1  Paris,  Gauthier-Villars  et  Fils,  1893. 

7 


98 


VECTOR  ANALYSIS 


FIG.  25. 


the  axis.  The  velocity  of  any  point  in  its  circle  is  equal 
to  the  product  of  the  angular  velocity  and  the  radius  of  the 
circle.  It  is  therefore  equal  to  the  product  of  the  angular 

velocity  and  the  perpendicular  dis- 
tance from  the  point  to  the  axis. 
The  direction  of  the  velocity  is 
perpendicular  to  the  axis  and  to 
the  radius  of  the  circle  described 
by  the  point. 

Let  a  (Fig.  25)  be  a  vector  drawn 
along  the  axis  of  rotation  in  that 
direction  in  which  a  right-handed 
screw  would  advance  if  turned  in 
the  direction  in  which  the  body  is 
rotating.  Let  the  magnitude  of  a 

be  a,  the  angular  velocity.  The  vector  a  may  be  taken  to 
represent  the  rotation  of  the  body.  Let  r  be  a  radius  vector 
drawn  from  any  point  of  the  axis  of  rotation  to  a  point  in  the 
body.  The  vector  product 

axr  =  a  r  sin  (a,  r) 

is  equal  in  magnitude  and  direction  to  the  velocity  v  of  the 
terminus  of  r.  For  its  direction  is  perpendicular  to  a  and  r 
and  its  magnitude  is  the  product  of  a  and  the  perpendicular 
distance  r  sin  (a,  r)  from  the  point  to  the  line  a.  That  is 

v  =  axr.  (45) 

If  the  body  be  rotating  simultaneously  about  several  axes 
a  a  •  •  •  which  pass  through  the  same  point  as   in  the 
the   velocities   due   to  the  various 


a 


2<  *3 


case  of  the  gyroscope 
rotations  are 


v.   = 


DIRECT  AND  SKEW  PRODUCTS  OF  VECTORS        99 

where  rx,  r2,  rg,  •  •  •  are  the  radii  vectores  drawn  from  points 
on  the  axis  a1}  a2,  a3,  •  •  •  to  the  same  point  of  the  body.  Let 
the  vectors  r1?  r2,  r3,  •  •  •  be  drawn  from  the  common  point  of 
intersection  of  the  axes.  Then 

rl  =  r2  =  r3  =  . .  • -  =  r 
and 

v  =  vx  +  v2  +  v3  +  •  •  •  =  axxr  +  a2xr  +  a3xr  H 

=  (a1  +  a2  +  a3  +  .  •  -)xr. 

This  shows  that  the  body  moves  as  if  rotating  with  the 
angular  velocity  which  is  the  vector  sum  of  the  angular 
velocities  a15  a2,  a3,  •  •  •  This  theorem  is  sometimes  known 
as  the  parallelogram  law  of  angular  velocities. 

It  will  be  shown  later  (Art.)  60  that  the  motion  of  any 
rigid  body  one  point  of  which  is  fixed  is  at  each  instant  of 
time  a  rotation  about  some  axis  drawn  through  that  point. 
This  axis  is  called  the  instantaneous  axis  of  rotation.  The 
axis  is  not  the  same  for  all  time,  but  constantly  changes  its 
position.  The  motion  of  a  rigid  body  one  point  of  which  is 
fixed  is  therefore  represented  by 

v  =  axr  (45) 

where  a  is  the  instantaneous  angular  velocity;  and  r,  the 
radius  vector  drawn  from  the  fixed  point  to  any  point  of  the 
body. 

The  most  general  motion  of  a  rigid  body  no  point  of  which 
is  fixed  may  be  treated  as  follows.  Choose  an  arbitrary 
point  0.  At  any  instant  this  point  will  have  a  velocity  v0. 
Eelative  to  the  point  0  the  body  will  have  a  motion  of  rotation 
about  some  axis  drawn  through  0.  Hence  the  velocity  v  of 
any  point  of  the  body  may  be  represented  by  the  sum  of 
v0  the  velocity  of  0  and  axr  the  velocity  of  that  point 

relative  to  0. 

v  =  v0  +  axr.  (46) 


100  VECTOR  ANALYSIS 

In  case  v0  is  parallel  to  a,  the  body  moves  around  a  and 
along  a  simultaneously.  This  is  precisely  the  motion  of  a 
screw  advancing  along  a.  In  case  v0  is  perpendicular  to  a,  it 
is  possible  to  find  a  point,  given  by  the  vector  r,  such  that 
its  velocity  is  zero.  That  is 

axr  =  —  v0. 

This  may  be  done  as  follows.     Multiply  by  xa. 

(axr)xa  —  —  v0xa 
or  a»a  r  —  a«r  a  =  —  v0xa. 

Let  r  be  chosen  perpendicular  to  a.     Then  a«r  is  zero  and 
a«a  r  =  —  v0  x  a 

-  ~  v°  x  a 
a*a 

The  point  r,  thus  determined,  has  the  property  that  its  veloc- 
ity is  zero.  If  a  line  be  drawn  through  this  point  parallel  to 
a,  the  motion  of  the  body  is  one  of  instantaneous  rotation 
about  this  new  axis. 

In  case  v0  is  neither  parallel  nor  perpendicular  to  a  it  may 
be  resolved  into  two  components 

V    =  V  '  4-  V  " 
vo  —  *o     i     vo 

which  are  respectively  parallel  and  perpendicular  to  a. 

v  =  v0'  4-  v0"  +  axr 
A  point  may  now  be  found  such  that 

v0"  =  —  axr. 

Let  the  different  points  of  the  body  referred  to  this  point  be 
denoted  by  r'.  Then  the  equation  becomes 

v  =  v0'  +  axr'.  (46)' 

The  motion  here  expressed  consists  of  rotation  about  an  axis 
a  and  translation  along  that  axis.  It  is  therefore  seen  that 
the  most  general  motion  of  a  rigid  body  is  at  any  instant 


DIRECT  AND  SKEW  PRODUCTS   OF   VECTORS      101 

the  motion  of  a  screw  advancing  at  a  certain  rate  along  a 
definite  axis  a  in  space.  The  axis  of  the  screw  and  its  rate 
of  advancing  per  unit  of  rotation  (i,  e.  its  pitch)  change  from 
instant  to  instant. 

52.]  The  conditions  for  equilibrium  as  obtained  by  the 
principle  of  virtual  velocities  may  be  treated  by  vector 
methods.  Suppose  any  system  of  forces  f  x,  f  2,  •  •  •  act  on  a 
rigid  body.  If  the  body  be  displaced  through  a  vector  dis- 
tance D  whether  this  distance  be  finite  or  infinitesimal  the 
work  done  by  the  forces  is 

D-fj,  D.fa,  ... 

The  total  work  done  is  therefore 

W  =  D.f  j  +  D.f  2  +  ••• 

If  the  body  be  in  equilibrium  under  the  action  of  the  forces 
the  work  done  must  be  zero. 

W  =  D-f  j  +  D-f  2  +  .  .  •  =  D«(f  !  +  f  2  +  •••)=  D.E  =  0. 

The  work  done  by  the  forces  is  equal  to  the  work  done  by 
their  resultant.  This  must  be  zero  for  every  displacement 
D.  The  equation 

D.R  =  O 

holds  for  all  vectors  D.    Hence 


The  total  resultant  must  be  zero  if  the  body  be  in  equilibrium. 

The  work  done  by  a  force  f  when  the  rigid  body  is  dis- 

placed by  a  rotation  of  angular  velocity  a  for  an  infinitesimal 

time  t  is  approximately 

a«dxf  t, 

where  d  is  a  vector  drawn  from  any  point  of  the  axis  of  rota- 
tion a  to  any  point  of  f.  To  prove  this  break  up  f  into  two 
components  f  ',  f  "  parallel  and  perpendicular  respectively  to  a. 

a-dxf  =  a-dxf  '  +  a-dxf  ". 


102  VECTOR  ANALYSIS 

As  £'  is  parallel  to  a  the  scalar  product  [a  d  f ']  vanishes. 
a»dxf  =  a.dxf  ". 

On  the  other  hand  the  work  done  by  f "  is  equal  to  the  work 
done  by  f  during  the  displacement.  For  f '  being  parallel  to 
a  is  perpendicular  to  its  line  of  action.  If  h  be  the  common 
vector  perpendicular  from  the  line  a  to  the  force  t",  the  work 
done  by  f "  during  a  rotation  of  angular  velocity  a  for  time 
t  is  approximately 

W=hf"  at  =  *hxt"t. 

The  vector  d  drawn  from  any  point  of  a  to  any  point  of  f  may 
be  broken  up  into  three  components  of  which  one  is  h,  another 
is  parallel  to  a,  and  the  third  is  parallel  to  f ".  In  the  scalar 
triple  product  [a  d  f "]  only  that  component  of  d  which  is 
perpendicular  alike  to  a  and  f "  has  any  effect.  Hence 

W=  a»hxf"  t  =  a.dxf  f  =  a»dxf  t. 

If  a  rigid  body  upon  which  the  forces  tlt  f  2,  •  •  •  act  be  dis- 
placed by  an  angular  velocity  a  for  an  infinitesimal  time  t 
and  if  d15  d2,  •  •  •  be  the  vectors  drawn  from  any  point  0  of 
a  to  any  points  of  f 1?  f 2,  •  •  •  respectively,  then  the  work  done 
by  the  forces  f  v  f2,  •  •  •  will  be  approximately 

W=  (a^xfj  +  a.d2xf2  +  •••)* 
=  a-CdjXfj  +  d2xf2  +  ..-)  t 
=  a.M0  {fj.fj,,  •••}  t. 

If  the  body  be  in  equilibrium  this  work  must  be  zero. 
Hence  a.M0  |flf  f2,  . . .}  t  =  0. 

The  scalar  product  of  the  angular  velocity  a  and  the  total 
moment  of  the  forces  tv  fa,  . . .  about  any  point  0  must  be 

zero.  As  a  may  be  any  vector  whatsoever  the  moment  itself 
must  vanish. 

Mo  {tv  tv  . . .}  =  0. 


DIRECT  AND  SKEW  PRODUCTS   OF   VECTORS      103 

The  necessary  conditions  that  a  rigid  body  be  in  equilib- 
rium under  the  action  of  a  system  of  forces  is  that  the  result- 
ant of  those  forces  and  the  total  moment  about  any  point  in 
space  shall  vanish. 

Conversely  if  the  resultant  of  a  system  of  forces  and  the 
moment  of  those  forces  about  any  one  particular  point  in  space 
vanish  simultaneously,  the  body  will  be  in  equilibrium. 

If  £  =  0,  then  for  any  displacement  of  translation  D 

D.R  =  0. 


and  the  total  work  done  is  zero,  when  the  body  suffers  any 
displacement  of  translation. 

Let  MO  {fp  f  2,  •  •  •  \  be  zero  for  a  given  point  0.  Then  for 
any  other  point  0' 

MO-  {fi?  f2>  •  •  'I  =  MO  {  f:,  f2,  •  •  •}  +  MO*  {BO}. 

But  by  hypothesis  R  is  also  zero.    Hence 

MO'  (fp  f  2»  •  •  •  I  —  0. 
Hence  a«Mo'  {fp  fa»  •  •  •  \  t  =  Q 

where  a  is  any  vector  whatsoever.  But  this  expression  is 
equal  to  the  work  done  by  the  forces  when  the  body  is  rotated 
for  a  time  t  with  an  angular  velocity  a  about  the  line  a 
passing  through  the  point  0'.  This  work  is  zero. 

Any  displacement  of  a  rigid  body  may  be  regarded  as  a 
translation  through  a  distance  D  combined  with  a  rotation 
for  a  time  t  with  angular  velocity  a  about  a  suitable  line  a  in 
space.  It  has  been  proved  that  the  total  work  done  by  the 
forces  during  this  displacement  is  zero.  Hence  the  forces 
must  be  in  equilibrium.  The  theorem  is  proved. 


104  VECTOR  ANALYSIS 

Applications  to  Geometry 

53.]  Relations  between  two  right-handed  systems  of  three 
mutually  perpendicular  unit  vectors.  —  Let  i,  j,  k  and  i',  j',  k' 
be  two  such  systems.  They  form  their  own  reciprocal  systems. 

Hence 

r  =  r.ii  +  r.jj  +  r-kk 

and  r  =  r.i'i'  +  r.j'j'  +  r.k'k'. 

From  this 


1 


i'  =  i'.i  i  +  i'.j  j  +  i'.k  k  =  ax  i  +  «2  j  +  «3  k 

j'   =  j'.i  i  +  j'.j  j  +  j'.k  k  =  b,  i  +  J2  j  +  53  k        (47') 

k'  =  k'.i  i  +  k'-j  j  +  k'.k  k  =  Cj  i  +  c2  j  +  c3  k. 


The  scalarsaj,  «2,  a3;  bv  Z>2,  b  3;  cr  c2,  c3  are  respectively  the 
direction  cosines  of  i';  j';  k'  with  respect  to  i,  j,  k. 
That  is 

ax  =  cos  (i',  i)    a2  =  cos  (i',  j)    «3  =  cos  (i',  k) 

&!  =  cos   (j',  i)     Z>2  =  cos  (j',  j)    Z>3  =  cos  (j',  k)     (48) 

cj  =  cos  (k',  i)    c2  =  cos  (k',  j)    c3  =  cos  (k',  k). 

In  the  same  manner 

s  i  ==  i.i'i'  +  i-j'j'  +  i.k'  k'=  ^  i'  +  ^  j'  +  Cj  k' 

]  j  =  j-i'i'  +  j-j'J'  +  j-k'k'  =  aa  i'  +  &a  j'  +  c2  k'    (47)" 

(  k  =  k.i'i'  +  k-j'j'  +  k-k'k'  =  as  i'  +  6,  j'  +  c3  k' 


and  )  j-j    -  1  =  a22  +  622  +  C22 

C  k.k  =  1  =  a32  +  &32  +  C32 

r  i'.j'  =  0  =  ai  6j  +  aa  62  +  a8  J3 

j'-k'  =  0  =  6lCl  +62C2  +  53C3  (50) 

I  k'.i'  =  0  =  cj  a,  +  c2  a2  +  c3  as 


DIRECT  AND  SKEW  PRODUCTS   OF   VECTORS      105 


and 


and 


i.j  =  0  =  a1  «2  +  &j  62  -f  Cj  c2 
j.k  =  0  =  «2  «3  +  &2  bs  +  c2  c3 


k'  =  i'xj'=  (aa  &g  - 


Cj       C2       Cg 


(50)' 


(51) 


i  +  («8  6j  -  ax  68)  j 


But 


k'  =  Cj  i  +  c2  j  4-  c3  k. 


Hence 


(52) 


Or 


Co      = 


and  similar  relations  may  be  found  for  the  other  six  quantities 
ax,  «2,  «3  ;  Jp  J2,  J3.  All  these  scalar  relations  between  the 
coefficients  of  a  transformation  which  expresses  one  set  of 
orthogonal  axes  JT',  Y\  Z1  in  terms  of  another  set  X,  I7",  Z  are 
important  and  well  known  to  students  of  Cartesian  methods. 
The  ease  with  which  they  are  obtained  here  may  be  note- 
worthy. 

A  number  of  vector  relations,  which  are  perhaps  not  so  well 
known,  but  nevertheless  important,  may  be  found  by  multi- 
plying the  equations 

i'  =  0   i  +  «      +  «   k 


in  vector  multiplication. 


-  a2k. 


(53) 


The  quantity  on  either  side  of  this  equality  is  a  vector.  From 
its  form  upon  the  right  it  is  seen  to  possess  no  component  in 


106  VECTOR  ANALYSIS 

the  i  direction  but  to  lie  wholly  in  the  jk-plane  ;  and  from 
its  form  upon  the  left  it  is  seen  to  lie  in  the  j'k  '-plane. 
Hence  it  must  be  the  line  of  intersection  of  those  two  planes. 
Its  magnitude  is  V  «a2  +  «32  or  V  bf  +  c^.  This  gives  the 
scalar  relations 


The  magnitude  1  —  af  is  the  square  of  the  sine  of  the  angle 
between  the  vectors  i  and  i'.  Hence  the  vector 

&1k'-c1j'  =  a,j-ask  (53) 

is  the  line  of  intersection  of  the  j'k'-  and  jk-planes,  and 
its  magnitude  is  the  sine  of  the  angle  between  the  planes. 
Eight  other  similar  vectors  may  be  found,  each  of  which  gives 
one  of  the  nine  lines  of  intersection  of  the  two  sets  of  mu- 
tually orthogonal  planes.  The  magnitude  of  the  vector  is  in 
each  case  the  sine  of  the  angle  between  the  planes. 

54.]  Various  examples  in  Plane  and  Solid  Geometry  may 
be  solved  by  means  of  products. 

Example  1  :  The  perpendiculars  from  the  vertices  of  a  trian- 
gle to  the  opposite  sides  meet  in  a  point.  Let  A  B  C  be  the 
triangle.  Let  the  perpendiculars  from  A  to  B  C  and  from  B 
to  CA  meet  in  the  point  0.  To  show  0  C  is  perpendicular 
to  A  B.  Choose  0  as  origin  and  let  OA  —  A,  OB  =  B,  and 
~C.  Then 


By  hypothesis 

A.(C  -  B)  =  0 

aQd  B-(A  -  C)  =  0. 

Subtract;  C.(B-A)  =  0, 

which  proves  the  theorem. 

Example  2:    To  find  the  vector  equation  of  a  line  drawn 
through  the  point  B  parallel  to  a  given  vector  A. 


DIRECT  AND  SKEW  PRODUCTS  OF   VECTORS      107 

Let  0  be  the  origin  and  B  the  vector  OB.  Let  E  be  the  ra- 
dius vector  from  0  to  any  point  of  the  required  line.  Then 
£  —  B  is  parallel  to  A.  Hence  the  vector  product  vanishes. 

Ax(R-B)  =  0. 

^\ 
This  is  the  desired  equation.     It  is  a  vector  equation  in  the 

unknown  vector  R.  The  equation  of  a  plane  was  seen  (page 
88)  to  be  a  scalar  equation  such  as 

E»C  =  c 
in  the  unknown  vector  B. 

The  point  of  intersection  of  a  line  and  a  plane  may  be 
found  at  once.  The  equations  are 

(  Ax(B  -  B)  =  0 
\         B.C  =  c 

AxB  =  AxB 

(AxR)xC  =  (AxB)xC 

A.C  B  -  C.R  A  =  (AxB)xC 

A.C  B  -  c  A  =  (AxB)xC 

Hence  (AxB)xC  +  c  A      , 

A.C 

The  solution  evidently  fails  when  A  •  C  =  0.  In  this  case  how- 
ever the  line  is  parallel  to  the  plane  and  there  is  no  solution ; 
or,  if  it  lies  in  the  plane,  there  are  an  infinite  number  of  solu- 
tions. 

Example  3:  The  introduction  of  vectors  to  represent  planes. 

Heretofore  vectors  have  been  used  to  denote  plane  areas  of 
definite  extent.  The  direction  of  the  vector  was  normal  to 
the  plane  and  the  magnitude  was  equal  to  the  area  to  be  re- 
presented. But  it  is  possible  to  use  vectors  to  denote  not  a 
plane  area  but  the  entire  plane  itself,  just  as  a  vector  represents 
a  point.  The  result  is  analogous  to  the  plane  coordinates  of 
analytic  geometry.  Let  0  be  an  assumed  origin.  Let  M N  be 
a  plane  in  space.  The  plane  M  N  is  to  be  denoted  by  a  vector 


108  VECTOR  ANALYSIS 

whose  direction  is  the  direction  of  the  perpendicular  dropped 
upon  the  plane  from  the  origin  0  and  whose  magnitude  is  the 
reciprocal  of  the  length  of  that  perpendicular.  Thus  the  nearer 
a  plane  is  to  the  origin  the  longer  will  be  the  vector  which 
represents  it. 

If  r  be  any  radius  vector  drawn  from  the  origin  to  a  point 
in  the  plane  and  if  p  be  the  vector  which  denotes  the  plane, 

then 

r.p  =  l 

is  the  equation  of  the  plane.     For 

r.p  =  r  cos  (r,  p)  p. 

Now  PI  the  length  of  p  is  the  reciprocal  of  the  perpendicular 
distance  from  0  to  the  plane.     On  the  other  hand  r  cos  (r,  p) 
is  that  perpendicular  distance.     Hence  r»p  must  be  unity. 
If  r  and  p  be  expressed  in  terms  of  i,  j,  k 


p  =  ui  +  vj  +  wk 
Hence  r»p  =  xu  +  yv  +  zw  =  l. 

The  quantities  u,  v,  w  are  the  reciprocals  of  the  intercepts  of 
the  plane  p  upon  the  axes. 

The  relation  between  r  and  p  is  symmetrical.     It  is  a  rela- 
tion of  duality.    If  in  the  equation 


r  be  regarded  as  variable,  the  equation  represents  a  plane  p 
which  is  the  locus  of  all  points  given  by  r.  If  however  p  be 
regarded  as  variable  and  r  as  constant,  the  equation  repre- 
sents a  point  r  through  which  all  the  planes  p  pass.  The 
development  of  the  idea  of  duality  will  not  be  carried  out 
It  is  familiar  to  all  students  of  geometry.  The  use  of  vec- 
tors to  denote  planes  will  scarcely  be  alluded  to  again  until 
Chapter  VII. 


DIRECT  AND  SKEW  PRODUCTS   OF  VECTORS      109 


SUMMARY  OF  CHAPTER  II 

The  scalar  products  of  two  vectors  is  equal  to  the  product 
of  their  lengths  multiplied  by  the  cosine  of  the  angle  between 

them. 

A.B  =  A  B  cos  (A,  B)  (1) 

A.B  =  B.A  (2) 

A»A  =  ^2.  (3) 

The  necessary  and  sufficient  condition  for  the  perpendicularity 
of  two  vectors  neither  of  which  vanishes  is  that  their  scalar 
product  vanishes.  The  scalar  products  of  the  vectors  i,  j,  k 

are 

M  =  j'j  =  k»k  =  1 

i.j  =j.k  =  k.i  =  0 

A.B  =  A1B1  +  A2  B2  +  As  B3  (7) 

A-A  =  A2  =  A?  +  A*  +  A*.  (8) 

If  the  projection  of  a  vector  B  upon  a  vector  A  is  B', 


The  vector  product  of  two  vectors  is  equal  in  magnitude  to 
the  product  of  their  lengths  multiplied  by  the  sine  of  the  an- 
gle between  them.  The  direction  of  the  vector  product  is  the 
normal  to  the  plane  of  the  two  vectors  on  that  side  on  which 
a  rotation  of  less  than  180°  from  the  first  vector  to  the  second 

appears  positive. 

AxB  =  A  B  sin  (A,  B)  c.  (9) 

The  vector  product  is  equal  in  magnitude  and  direction  to  the 
vector  which  represents  the  parallelogram  of  which  A  and  B 
are  the  two  adjacent  sides.  The  necessary  and  sufficient  con- 
dition for  the  parallelism  of  two  vectors  neither  of  which 


110 


VECTOR  ANALYSIS 


vanishes  is   that  their  vector  product  vanishes.     The  com- 
mutative laws  do  not  hold. 


AxB  = 


Bs  - 


AxB=-BxA 
ixi    =  jxj  =  kxk  = 
ixj    =—  jxi  =  k 
jxk  =  —  kxj  =  i 
kxi  =  —  ixk  =  j 
£a)  i  +  (^3  B1  -  Al 


(10) 
(12) 


AxB  = 


j 

A, 


(13) 
(13)' 


The  scalar  triple  product  of  three  vectors  [A  B  C]  is  equal 
to  the  volume  of  the  parallelepiped  of  which  A,  B,  C  are  three 
edges  which  meet  in  a  point. 


(15)' 


[AB  C]  =  A-BxC  =  B.CxA  =  C«AxB 
=  AxB»C  =  BxC-A  =  CxA-B 
[ABC]  =-  [ACB].  (16)' 


The  dot  and  the  cross  in  a  scalar  triple  product  may  be  inter- 
changed and  the  order  of  the  letters  may  be  permuted  cyclicly 
without  altering  the  value  of  the  product ;  but  a  change  of 
cyclic  order  changes  the  sign. 


[ABC]  = 


(18)' 


[ABC]  = 


[a  be] 


(19)' 


DIRECT  AND  SKEW  PRODUCTS   OF  VECTORS      111 
If  the  component  of  B  perpendicular  to  A  be  B", 

>_  (20) 

« 

Ax (BxC)  =  A.C  B  -  A«B  C  (24) 

(AxB)xC  =  A-C  B  -  C.B  A  (24)' 

(AxB).(CxD)  =  A.C  B.D  -  A.D  B.C  (25) 
(AxB)x(CxD)  =  [A CD]  B  -  [BCD]  A 

=  [ABD]  C-[ABC]  D.  (26) 

The  equation  which  subsists  between  four  vectors  A,  B,  C,  D 
is 

[BCD]  A-[CDA]B+  [DAB]  C-  [ABC]  D  =  0.      (27) 

Application  of  formulae  of  vector  analysis  to  obtain  the  for- 
mulae of  Plane  and  Spherical  Trigonometry. 

The  system  of  vectors  a',  V,  c'  is  said  to  be  reciprocal  to  the 
system  of  three  non-coplanar  vectors  a,  b,  c 

.      bxc  cxa  axb 

when  a'=  ,..-,»    V  =  p-— r'    c'  =  — — •      (29) 

[a  be]  [a  b  c]  [a  b  c] 

A  vector  r  may  be  expressed  in  terms  of  a  set  of  vectors  and 
its  reciprocal  in  two  similar  ways 

r  =  r.a'  a  +  r.b'  b  +  r.c'  c  (30) 

r  =  r.aa'  +  r.bb'  +  r.cc'.  (31) 

The  necessary  and  sufficient  conditions  that  the  two  systems  of 
non-coplanar  vectors  a,  b,  c  and  a',  b',  c'  be  reciprocals  is  that 

a'.a  =  b'.b  =  c'«c  =  1 
a'.b  =  a'.c  =  b'.c  =  b'.a  =  c'-a  =  c'-b  =  0. 

If  af, b',  c'  form  a  system  reciprocal  to  a,  b,  c ;  then  a,  b,  c  will 
form  a  system  reciprocal  to  a',  b',  c'. 


112  VECTOR  ANALYSIS 


[PftR]  [ABC]- 


P.A     P.B     P.C 

Q.A  a-B   a-c 
E.A   R.B   R.C 


(34) 


The  system  i,  j,  k  is  its  own  reciprocal  and  if  conversely  a 
system  be  its  own  reciprocal  it  must  be  a  right  or  left  handed 
system  of  three  mutually  perpendicular  unit  vectors.  Appli- 
cation of  the  theory  of  reciprocal  systems  to  the  solution  of 
scalar  and  vector  equations  of  the  first  degree  in  an  unknown 
vector.  The  vector  equation  of  a  plane  is 

r-A  =  a.  (36) 

Applications  of  the  methods  developed  in  Chapter  II.,  to  the 
treatment  of  a  system  of  forces  acting  on  a  rigid  body  and  in 
particular  to  the  reduction  of  any  system  of  forces  to  a  single 
force  and  a  couple  of  which  the  plane  is  perpendicular  to  that 
force.  Application  of  the  methods  to  the  treatment  of 
instantaneous  motion  of  a  rigid  body  obtaining 

v  =  v0  +  a  x  r  (46) 

where  v  is  the  velocity  of  any  point,  v0  a  translational  veloc- 
ity in  the  direction  a,  and  a  the  vector  angular  velocity  of  ro- 
tation. Further  application  of  the  methods  to  obtain  the 
conditions  for  equilibrium  by  making  use  of  the  principle  of 
virtual  velocities.  Applications  of  the  method  to  obtain 
the  relations  which  exist  between  the  nine  direction  cosines 
of  the  angles  between  two  systems  of  mutually  orthogonal 
axes.  Application  to  special  problems  in  geometry  including 
the  form  under  which  plane  coordinates  make  their  appear- 
ance in  vector  analysis  and  the  method  by  which  planes  (as 
distinguished  from  finite  plane  areas)  may  be  represented 
by  vectors. 


DIRECT  AND   SKEW  PRODUCTS   OF   VECTORS      113 

EXERCISES  ON  CHAPTER  II 
Prove  the  following  reduction  formulae 

1.  Ax{Bx(CxD)}  =  [ACD]B-  A-B  CxD 

=  B.D  AxC  -  B-C  AxD. 

2.  [AxB  CxD  ExF]  =  [A  B  D]  [C  E  F]  -  [A  B  C]  [D  E  F] 

=  [ABE]  [FCD]  -  [ABF]  [E  C  D] 
=  [CD A]  [BEF]  -  [CDB]  [AEFj. 

3.  [AxB    BxC     CxA]  =  [ABC]2. 


4.    [PQR]  (AxB)  = 


P.A    P.B 

Q-A     Q.B 


R-A    R.B    R 

5.  Ax(BxC)  +  Bx(CxA)  +  Cx(AxB)  =  0. 

6.  [AxP    BxQ    CxR]  +  [AxQ    BxR    CxP] 

+  [AxR    BxP    CxQ]  =  0. 

7.  Obtain  formula  (34)  in  the  text  by  expanding 

[(AxB)xP].[Cx(CtxR)] 
in  two  different  ways  and  equating  the  results. 

8.  Demonstrate   directly  by  the  above   formulae    that  if 
a',  b',  c'  form  a  reciprocal  system  to  a,  b,  c;  then  a,  b,  c  form 
a  system  reciprocal  to  a',  b',  c'. 

9.  Show  the  connection  between  reciprocal  systems  of  vec- 
tors and  polar  triangles  upon  a  sphere.     Obtain  some  of  the 
geometrical  formulae  connected  with  polar  triangles  by  inter- 
preting vector  formulae  such  as  (3)  in  the  above  list. 

10.  The  perpendicular  bisectors  of  the  sides  of  a  triangle 
meet  in  a  point. 

11.  Find  an  expression  for  the  common  perpendicular  to 
two  lines  not  lying  in  the  same  plane. 


114 


VECTOR  ANALYSIS 


12.  Show  by  vector  methods  that  the  formulae  for  the  vol- 
ume of  a  tetrahedron  whose  four  vertices  are 


is 


1/3 


13.   Making  use  of  formula  (34)  of  the  text  show  that 


[a  b  c]  =  a  b  c 


N 


1  n m 
n  1  I 
m  I  1 


where  a,  b,  c  are  the  lengths  of  a,  b,  c  respectively  and  where 
I  =  cos  (b,  c),    m  =  cos  (c,  a),    n  =  cos  (a,  b). 

14.  Determine    the   perpendicular  (as  a   vector  quantity) 
which  is  dropped  from  the  origin  upon  a  plane  determined  by 
the  termini  of  the  vectors  a,  b,  c.     Use  the  method  of  solution 
given  in  Art.  46. 

15.  Show  that  the  volume  of  a  tetrahedron  is  equal  to  one 
sixth  of  the  product  of  two  opposite  edges  by  the  perpendicu- 
lar distance  between  them  and  the  sine  of  the  included  angle. 

16.  If  a  line  is  drawn  in  each  face  plane  of  any  triedral  angle 
through  the  vertex  and  perpendicular  to  the  third  edge,  the 
three  lines  thus  obtained  lie  in  a  plane. 


CHAPTER  III 


THE  DIFFEBENTIAL   CALCULUS   OF  VECTOES 

Differentiation  of  Functions  of  One  Scalar  Variable 

55.]  IF  a  vector  varies  and  changes  from  r  to  r'  the  incre- 
ment of  r  will  be  the  difference  between  r'  and  r  and  will  be 
denoted  as  usual  by  A  r. 

Ar  =  r'-r,  (1) 

where  A  r  must  be  a  vector  quantity.  If  the  variable  r  be 
unrestricted  the  increment  A  r  is  of  course  also  unrestricted : 
it  may  have  any  magnitude  and  any  direction.  If,  however, 
the  vector  r  be  regarded  as  a  function  (a  vector  function)  of 
a  single  scalar  variable  t  the  value  of  A  r  will  be  completely 
determined  when  the  two  values  t  and  t'  of  £,  which  give  the 
two  values  r  and  r',  are  known. 

To  obtain  a  clearer  conception  of  the  quantities  involved 
it  will  be  advantageous  to  think  of  the  vector  r  as  drawn 
from  a  fixed  origin  0  (Fig.  26).  When 
the  independent  variable  t  changes  its 
value  the  vector  r  will  change,  and  as  t 
possesses  one  degree  of  freedom  r  will 
vary  in  such  a  way  that  its  terminus 
describes  a  curve  in  space,  r  will  be 
the  radius  vector  of  one  point  P  of 
the  curve;  r',  of  a  neighboring  point  P'. 
chord  P  P'  of  the  curve.  The  ratio 


O       FIG.  26. 
A  r  will  be  the 


Ar 

Al 


VECTOR  ANALYSIS 

will  be  a  vector  collinear  with  the  chord  PP'  but  magnified 
in  the  ratio  1 :  A  t.  When  A  t  approaches  zero  P'  will  ap- 
proach P,  the  chord  PP7  will  approach  the  tangent  at  P,  and 

the  vector 

Ar  ...                ,    dT 

-  will  approach  — 

A  t  dt 

which  is  a  vector  tangent  to  the  curve  at  P  directed  in  that 
sense  in  which  the  variable  t  increases  along  the  curve. 
If  r  be  expressed  in  terms  of  i,  j,  k  as 


the  components  rv  r2,  rz  will  be  functions  of  the  scalar  t. 
r'  =  (>!  +  A  rt)  i  +  O2  +  A  ra)  j  +  (r8  +  A  r3)  k 


A  r      A  r,  .       A  r2  .       A  n  . 

—          l  i    i        $  i  4-  -   -  V 

T       A      J      J    ^       A.I       * 


^      A*         A*  '       A* 
t?  r      c 


and 


Hence  the  components  of  the  first  derivative  of  r  with  re- 
spect to  t  are  the  first  derivatives  with  respect  to  t  of  the 
components  of  r.  The  same  is  true  for  the  second  and  higher 
derivatives. 


. 
_i  __    \  j  __    |_ 

*  k' 


_      _  __        __ 

dt*      dt*          dt*          dt* 

(2)' 
dn  r      d*  rt  .      dnr2  .      dnr3 

d  t»  ~  d  t*  l  r  7r  J  +  ~dr'  k' 

In  a  similar  manner  if  r  be  expressed  in  terms  of  any  three 
non-coplanar  vectors  a,  b,  c  as 

r  —  aa  +  5b  +  cc 
d^r_d^        <Z*J        dnc 

"~a  +         *c' 


THE  DIFFERENTIAL   CALCULUS   OF  VECTORS     117 

Example  :     Let          r  =  a  cos  t  +  b  sin  t. 

The  vector  r  will  then  describe  an  ellipse  of  which  a  and  b 
are  two  conjugate  diameters.  This  may  be  seen  by  assum- 
ing a  set  of  oblique  Cartesian  axes  X,  Y  coincident  with  a 

and  b.     Then 

X  =  a  cos  t,       Y=b  sin  t, 


which  is  the  equation  of  an  ellipse  referred  to  a  pair  of  con- 
jugate diameters  of  lengths  a  and  5  respectively. 

dr 

—  —  =  —  a  sin  t  +  b  cos  t. 

a  t 

Hence          -  =  a  cos  (  t  +  90°)  +  b  sin  (t  +  90°). 

CL  t 

The  tangent  to  the  curve  is  parallel  to  the  radius  vector 

for  0  +  90°).         2 

=  —  (a  cos  t  +  b  sin  t). 


The  second  derivative  is  the  negative  of  r.     Hence 

_ 

~ 


is  evidently  a  differential  equation  satisfied  by  the  ellipse. 

Example  :    Let          r  =  a  cosh  t  +  b  sinh  t. 
The  vector  r  will  then  describe  an  hyperbola  of  which  a  and 
b  are  two  conjugate  diameters. 

dr 

—  =  a  sinh  t  +  b  cosh  t, 

CL   b 


and  -  =  a  cosh  t  +  b  sinh  t. 


TT 

Hence  —  —  -  —  r 

dt2 

is  a  differential  equation  satisfied  by  the  hyperbola. 


118  VECTOR  ANALYSIS 

56  ]  A  combination  of  vectors  all  of  which  depend  on  the 
same  scalar  variable  t  may  be  differentiated  very  much  as  in 
ordinary  calculus. 


For 

(a 


A  (a  •  b)  =  (a  +  A  a)  •  (b  +  Ab)  -  a  •  b 


A(a.b)  Ab      Aa  Aa-Ab 

— -  =  a  •  TT  +  ~T7  '  b  +  A~7 

A2  A£        A£  Ac 

Hence  in  the  limit  when  A  t  =  0, 

d  db     da.    ,  ,o, 
(a»b)  =  a»-r — f-  —  •  b                       (o) 

<^  /  d  b\       /  cZ  a\       .  //<x 

-(axb)  =  ax  l77J+(T7)  xb  <4> 


d  fdv\ 

—  (a«bxc)  =  a.bx  I      -\  +  a«(  ^-Ixc 

dt v  ' 


(5) 


[bx.].  (6) 


The  last  three  of  these  formulae  may  be  demonstrated  exactly 
as  the  first  was. 

The  formal  process  of  differentiation  in  vector  analysis 
differs  in  no  way  from  that  in  scalar  analysis  except  in  this 
one  point  in  which  vector  analysis  always  differs  from  scalar 
analysis,  namely  :  The  order  of  the  factors  in  a  vector  product 


THE  DIFFERENTIAL    CALCULUS   OF    VECTORS      119 

cannot  be  changed  without  changing  the  sign  of  the  product. 
Hence  of  the  two  formulae 


d 


and 


the  first  is  evidently  incorrect,  but  the  second  correct.  In 
other  words,  scalar  differentiation  must  take  place  without 
altering  the  order  of  the  factors  of  a  vector  product.  The 
factors  must  be  differentiated  in  situ.  This  of  course  was  to 
be  expected. 

In  case  the  vectors  depend  upon  more  than  one  variable 
the  results  are  practically  the  same.  In  place  of  total  deriva- 
tives with  respect  to  the  scalar  variables,  partial  derivatives 
occur.  Suppose  a  and  b  are  two  vectors  which  depend  on 
three  scalar  variables  #,  y,  z.  The  scalar  product  a«b  will 
depend  upon  these  three  variables,  and  it  will  have  three 
partial  derivatives  of  the  first  order. 


CT) 


The  second  partial  derivatives  are  formed  in  the  same  way. 


5x9y 


120  VECTOR   ANALYSIS 

Often  it  is  more  convenient  to  use  not  the  derivatives  but 
the  differentials.  This  is  particularly  true  when  dealing  with 
first  differentials.  The  formulae  (3),  (4)  become 

d  (a-b)  =  da.b  +  a  •  db,  (3)' 

d  (a  X  b)  =  da.  x  b  +  a  x  db,  (4)' 

and  so   forth.     As  an  illustration    consider   the  following 
example.     If  r  be  a  unit  vector 


The  locus  of  the  terminus  of  r  is  a  spherical  surface  of  unit 
radius  described  about  the  origin,  r  depends  upon  two  vari- 
ables. Differentiate  the  equation. 

(d  r)  •  r  +  r  •  (d  r)  =  2  r  .  (d  r)  =  0. 
Hence  r  •  d  r  =  0. 

Hence  the  increment  dr  of  a  unit  vector  is  perpendicular  to 
the  vector.  This  can  be  seen  geometrically.  If  r  traces  a 
sphere  the  variation  d  r  must  be  at  each  point  in  the  tangent 
plane  and  hence  perpendicular  to  r. 

*57.]  Vector  methods  may  be  employed  advantageously 
in  the  discussion  of  curvature  and  torsion  of  curves.  Let  r 
denote  the  radius  vector  of  a  curve 


where  f  is  some  vector  function  of  the  scalar  t.  In  most  appli- 
cations in  physics  and  mechanics  t  represents  the  time.  Let 
s  be  the  length  of  arc  measured  from  some  definite  point  of 
the  curve  as  origin.  The  increment  A  r  is  the  chord  of  the 
curve.  Hence  A  r  /  A  s  is  approximately  equal  in  magnitude 
to  unity  and  approaches  unity  as  its  limit  when  A  s  becomes 
infinitesimal.  Hence  dr/ds  will  be  a  unit  vector  tangent 
the  curve  and  will  be  directed  toward  that  portion  of  the 


THE  DIFFERENTIAL   CALCULUS   OF   VECTORS      121 


curve  along  which  s  is  increasing  (Fig.  27).     Let  t  be  the 
unit  tangent 

£  =  t  (8)          , 


The  curvature  of  the  curve  is  the 
limit  of  the  ratio  of  the  angle  through 
which  the  tangent  turns^  to  tile  length 

of  the  arc.  The  tangent  changes  by  the  increment  At.  As  t 
is  of  unit  length,  the  length  of  A  t  is  approximately  the  angle 
through  which  the  tangent  has  turned  measured  in  circular 
measure.  Hence  the  directed  curvature  C  is 

d*r 


C  = 


LIM 


At 


As=0   As 


dt 
ds 


ds* 


The  vector  C  is  collinear  with  A  t  and  hence  perpendicular  to 
t ;  for  inasmuch  as  t  is  a  unit  vector  A  t  is  perpendicular 
to  t. 

The  tortuosity  of  a  curve  is  the  limit  of  the  ratio  of  the 
angle  through  which  the  osculating  plane  turns  to  the  length 
of  the  arc.  The  osculating  plane  is  the  plane  of  the  tangent 
vector  t  and  the  curvature  vector  C.  The  normal  to  this 

planeis  N  =  txC. 

If  c  be  a  unit  vector  collinear  with  C 

n  =  t  x  c 

will  be  a  unit  normal  (Fig.  28)  to  the  osculating  plane  and 

the  three  vectors  t,  c,  n  form  an  i,  j,  k  system, 

that  is,  a  right-handed  rectangular  system. 

Then  the  angle  through  which  the  osculating 

plane  turns  will  be  given  approximately  by 

A  n  and  hence  the  tortuosity  is  by  definition 

d  n  /  d  s. 

From  the  fact  that  t,  c,  n  form  an  i,  j,  k  system  of  unit 
vectors 


FIG.  28. 


122  VECTOR  ANALYSIS 

t.t  =  c«c  =  n»n  =  l 
and  t«c  =  c»n  =  n»t  =  0. 

Differentiating  the  first  set 

t.dt  =  c«dc  =  n»d!n  =  0, 
and  the  second 

t»  do  +  dt'C  =  c»dn  +  dc»n  =  n«dt  +  dn»t  =  0. 
But  d  t  is  parallel  to  c  and  consequently  perpendicular  to  n. 

n»  dt  =  0. 

Hence  d  n  •  t  =  0. 

The  increment  of  n  is  perpendicular  to  t.  But  the  increment 
of  n  is  also  perpendicular  to  n.  It  is  therefore  parallel  to  c. 
As  the  tortuosity  is  T  =  dn/ds,  it  is  parallel  to  dn  and  hence 
too. 

The  tortuosity  T  is 

d  ,  d  fdr     d*r       1     \ 

T  =  —  (txc)  =  — (— x  —2     . )       (11) 

ds  dsds      ds* 


dr      d*r  d 

T  T~ 


s      dsz  ds  ^  c  »C 

The  first  term  of  this  expression  vanishes.  T  moreover  has 
been  seen  to  be  parallel  to  C  =  dz  i/d  s2.  Consequently  the 
magnitude  of  T  is  the  scalar  product  of  T  by  the  unit  vec- 
tor c  in  the  direction  of  C.  It  is  desirable  however  to  have 
the  tortuosity  positive  when  the  normal  n  appears  to  turn  in 
the  positive  or  counterclockwise  direction  if  viewed  from 
that  side  of  the  n  c-plane  upon  which  t  or  the  positive  part 
of  the  curve  lies.  With  this  convention  d  n  appears  to  move 
in  the  direction  —  c  when  the  tortuosity  is  positive,  that  is,  n 
turns  away  from  c.  The  scalar  value  of  the  tortuosity  will 
therefore  be  given  by  —  c .  T. 


THE  DIFFERENTIAL   CALCULUS  OF  VECTORS       123 

dr     dBr      1  dr      dzr  d       1 

—  c»T  =  —  c.  —  x  — „  — — —  —  c 


ds      ds3«Y/c«C  ^s      ^s2^sVC»C 

But  c  is  parallel  to  the  vector  d2  r/d  s2.     Hence 

c  •  -T-  X  -^— =  =  0. 


And  c  is  a  unit  vector  in  the  direction  C.     Hence 

C        dzr        1 


c  = 


Hence      r=-c.T  =  --— '•^x-^  — 

as2    as      ds*  C  •  C 




ft  ^     ct  *?        d  K 

ur  T  = (13) 


The  tortuosity  may  be  obtained  by  another  method  which 
is  somewhat  shorter  if  not  quite  so  straightforward. 

t»c  =  c«n  =  n«t  =  0. 
Hence  dt»c  =  —  dc  »t 

dc  «n  =  —  dn»c 
dn»t  =  —  dt*n. 

Now  d  t  is  parallel  to  c ;  hence  perpendicular  to  n.  Hence 
d  t  •  n  =  0.  Hence  dn»t  =  0.  But<Znis  perpendicular  to  n. 
Hence  d  n  must  be  parallel  to  c.  The  tortuosity  is  the  mag- 
nitude of  dn/ds  taken  however  with  the  negative  sign 
because  d  n  appears  clockwise  from  the  positive  direction  of 
the  curve.  Hence  the  scalar  tortuosity  T  may  be  given  by 

dn  dc 

r=-  —  .c  =  n.— ,  (14) 

ds  ds 

dp 

T  =  txc-^>  (14)' 

as 


124 


VECTOR  ANALYSIS 

C 

c  =  — — —  > 


do 

ds 


dC 


d 


c.c 


But 


dc  dC     ,——-. 

.  —  =  t  x  c  •  -  -  VC  -C  —  t  x  c 

ds  d  8 

t  x  c  •  C  =  0. 

dV    /^— « 
t  x  c  .  —  VC  •  C 

r= ^1 , 

c.c 

t  x  C  •  — 

d  s 

^TC— ' 


m 


T= 


dr 


dsz 


In  Cartesian  coordinates  this  becomes 

d  x      dy      dz 
ds       d  s       ds 


T= 


ds2  ds2  dsz 
ds  x  ds  y  dBz 
ds*  ds9  ds9 


ds2' 


(13) 


(13)' 


Those  who  would  pursue  the  study  of  twisted  curves  and 

surfaces  in  space  further  from  the  standpoint  of  vectors  will 

find  the  book  "  Application  de  la  Methode  Vectorielle  de  Grass- 

mann  d  la  Geomttrie   Infinitesimale"1  by  FEHB   extremely 

1  Paris,  Cam?  et  Naud,  1899. 


THE  DIFFERENTIAL   CALCULUS  OF   VECTORS      125 

helpful.  He  works  with  vectors  constantly.  The  treatment 
is  elegant.  The  notation  used  is  however  slightly  different 
from  that  used  by  the  present  writer.  The  fundamental 
points  of  difference  are  exhibited  in  this  table 

EJ  •  a  2  ~  [«j  «2] 

ct -I      *     Clo      X     Clo I   <*  1      <*  o     «•  Q    I     /N^     I  CZl      $o     ^Q    I  • 

1  Jj  O  L.       J.  £i  O_|  Li          i          o_| 

One  used  to  either  method  need  have  no  difficulty  with  the 
other.  All  the  important  elementary  properties  of  curves 
and  surfaces  are  there  treated.  They  will  not  be  taken 
up  here. 

*  Kinematics 

58.]  Let  r  be  a  radius  vector  drawn  from  a  fixed  origin  to 
a  moving  point  or  particle.  Let  t  be  the  time.  The  equation 
of  the  path  is  then 

The  velocity  of  the  particle  is  its  rate  of  change  of  position. 
This  is  the  limit  of  the  increment  A  r  to  the  increment  A  t. 

LIM    rArn      di 

*asA*-*.-nrTTL*jFr  (15) 


This  velocity  is  a  vector  quantity.  Its  direction  is  the 
direction  of  •  the  tangent  of  the  curve  described  by  the  par- 
ticle. The  term  speed  is  used  frequently  to  denote  merely 
the  scalar  value  of  the  velocity.  This  convention  will  be 
followed  here.  Then 


if  s  be  the  length  of  the  arc  measured  from  some  fixed  point 
of  the  curve.  It  is  found  convenient  in  mechanics  to  denote 
differentiations  with  respect  to  the  time  by  dots  placed  over 
the  quantity  differentiated.  This  is  the  oldfluxional  notation 


126  VECTOR  ANALYSIS 

introduced  by  Newton.    It  will  also  be  convenient  to  denote 
the  unit  tangent  to  the  curve  by  t.     The  equations  become 

dr 


T=r=j* 

•         dS  rtR\ 

v  =  s=-  (16) 

v  =  v  t.  (17) 

The  acceleration  is  the  rate  of  change  of  velocity.  It 
is  a  vector  quantity.  Let  it  be  denoted  by  A.  Then  by 
definition 

LIM     A  v     d  v      • 


_ 
=  =        = 


and 


Differentiate  the  expression        v  =  v  t. 

_dv_d(vt)      dv  dt 

=  d7  =  ~d~T  =  'dti^  V  di* 
d  v      dz  s      •• 


dt  _dt    d  s 
d  t      d  s    d  t 

where  C  is  the  (vector)  curvature  of  the  curve  and  v  is  the 
speed  in  the  curve.  Substituting  these  values  in  the  equation 
the  result  is  g?  ~  g  +  oZ 

A  =  s  t  +  0a  C. 

'  ^  %* 

The  acceleration  of  a  particle  moving  in  a  curve  has  there- 
fore been  broken  up  into  two  components  of  which  one  is  paral- 
lel to  the  tangent  t  and  of  which  the  other  is  parallel  to  the 
curvature  C,  that  is,  perpendicular  to  the  tangent.  That  this 
resolution  has  been  accomplished  would  be  unimportant  were 


THE  DIFFERENTIAL   CALCULUS  OF  VECTORS      127 

it  not  for  the  remarkable  fact  which  it  brings  to  light.  The 
component  of  the  acceleration  parallel  to  the  tangent  is  equal 
in  magnitude  to  the  rate  of  change  of  speed.  It  is  entirely 
independent  of  what  sort  of  curve  the  particle  is  describing. 
It  would  be  the  same  if  the  particle  described  a  right  line 
with  the  same  speed  as  it  describes  the  curve.  On  the  other 
hand  the  component  of  the  acceleration  normal  to  the  tangent 
is  equal  in  magnitude  to  the  product  of  the  square  of  the 
speed  of  the  particle  and  the  curvature  of  the  curve.  The 
sharper  the  curve,  the  greater  this  component.  The  greater 
the  speed  of  the  particle,  the  greater  the  component.  But  the 
rate  of  change  of  speed  in  path  has  no  effect  at  all  on  this 
normal  component  of  the  acceleration. 
If  r  be  expressed  in  terms  of  i,  j,  k  as 

r  =  #  i  +  yj  +  2  k, 
v  =  T  =  xi  +  j/j  +  zk,  (15)' 


v  —  V  x*  +  y*  +  «a,  (16)' 

A  =  v  =  r  =  »i  +  ^j+sk,  (18)' 

x  x  +  y  y  +  2  z      \V*v\ 
A  =  v=s=  y  ;a  ==g-    -^-       v 

^t  T  Jr    T 

From  these  formulae  the  difference  between  s>  the  rate  of 
change  of  speed,  and  A  =  f ,  the  rate  of  change  of  velocity, 
is  apparent.  Just  when  this  difference  first  became  clearly 
recognized  would  be  hard  to  say.  But  certain  it  is  that 
Newton  must  have  had  it  in  mind  when  he  stated  his  second 
law  of  motion.  The  rate  of  change  of  velocity  is  proportional 
to  the  impressed  force ;  but  rate  of  change  of  speed  is  not. 

59.]  The  hodograph  was  introduced  by  Hamilton  as  an 
aid  to  the  study  of  the  curvilinear  motion  of  a  particle. 
With  any  assumed  origin  the  vector  velocity  f  is  laid  off. 
The  locus  of  its  terminus  is  the  hodograph.  In  other  words, 
the  radius  vector  in  the  hodograph  gives  the  velocity  of  the 


128  VECTOR  ANALYSIS 

particle  in  magnitude  and  direction  at  any  instant.  It  is 
possible  to  proceed  one  step  further' and  construct  the  hodo- 
graph  of  the  hodograph.  This  is  done  by  laying  off  the 
vector  acceleration  A  =  r  from  an  assumed  origin.  The 
radius  vector  in  the  hodograph  of  the  hodograph  therefore 
gives  the  acceleration  at  each  instant. 

Example  1 :    Let  a  particle  revolve  in  a  circle  (Fig.  29) 

of  radius  r  with  a  uniform 
angular  velocity  a.  The 
speed  of  the  particle  will  then 
be  equal  to 

v  =  a  r. 

Let  r  be   the   radius   vector 
drawn  to  the  particle.     The 
velocity  v  is  perpendicular  to  r  and  to  a.     It  is 
r  =  v  =  a  x  r. 

The  vector  v  is  always  perpendicular  and  of  constant  magni- 
tude. The  hodograph  is  therefore  a  circle  of  radius  v  =  a  r. 
The  radius  vector  f  in  this  circle  is  just  ninety  degrees  in 
advance  of  the  radius  vector  r  in  its  circle,  and  it  conse- 
quently describes  the  circle  with  the  same  angular  velocity 
a.  The  acceleration  A  which  is  the  rate  of  change  of  v  is 
always  perpendicular  to  v  and  equal  in  magnitude  to 

A  =  a  v  =  a2  r. 

The  acceleration  A  may  be  given  by  the  formula 
r  =  A  =  axv  =  ax(axr)  =  a«r  a  —  a»a  r. 

But  as  a  is  perpendicular  to  the  plane  in  which  r  lies,  a  •  r  =  0. 

Hence 

r  =  A  =  —  a»a  r  =  —  a2  r. 

The  acceleration  due  to  the  uniform  motion  of  a  particle  in 
a  circle  is  directed  toward  the  centre  and  is  equal  in  magni- 
tude to  the  square  of  the  angular  velocity  multiplied  by  the 
radius  of  the  circle. 


THE  DIFFERENTIAL   CALCULUS  OF  VECTORS     129 

Example  2:    Consider  the  motion  of  a  projectile.     The 
acceleration  in  this  case  is  the  acceleration  g  due  to  gravity. 


The  hodograph  of  the  hodograph  reduces  to  a  constant 
vector.  The  curve  is  merely  a  point.  It  is  easy  to  find 
the  hodograph.  Let  v0  be  the  velocity  of  the  projectile 
in  path  at  any  given  instant.  At  a  later  instant  the  velocity 

will  be 

v  =  v0  +  t  g. 

Thus  the  hodograph  is  a  straight  line  parallel  to  g  and  pass- 
ing through   the   extremity   of  v0.      The    hodograph   of    a 
particle  moving  under  the  influence  of  gravity  is  hence  a 
straight  line.     The  path  is  well  known  to  be  a  parabola. 
Example  3  :     In  case  a  particle  move  under  any  central 

acceleration 

r  =  A  =  f(r). 

The  tangents  to  the  hodograph  of  r  are  the  accelerations  r. 
But  these  tangents  are  approximately  collinear  with  the 
chords  between  two  successive  values  r  and  f0  of  the  radius 
vector  in  the  hodograph.  That  is  approximately 


A* 
Multiply  by  rx.  r  X  r  =  r  x  — - — — . 

Since  r  and  r  are  parallel 

r  x  (r  -  r0)  =  0. 
Hence  r  x  r  =  r  x  f0 . 

But  ^  r  x  r  is  the  rate  of  description  of  area.  Hence  the 
equation  states  that  when  a  particle  moves  under  an  ac- 
celeration directed  towards  the  centre,  equal  areas  are  swept 
over  in  equal  times  by  the  radius  vector. 

9 


130  VECTOR  ANALYSIS 

Perhaps  it  would  be  well  to  go  a  little  more  carefully  into 
this  question.  If  r  be  the  radius  vector  of  the  particle  in 
its  path  at  one  instant,  the  radius  vector  at  the  next  instant 
is  r  +  A  r.  The  area  of  the  vector  of  which  r  and  r  +  A  r  are 
the  bounding  radii  is  approximately  equal  to  the  area  of  the 
triangle  enclosed  by  r,  r  +  A  r,  and  the  chord  A  r.  This 
area  is 

|rx(r  +  Ar)=irxr  +  ^r  xAr=|rxAr. 

The  rate  of  description  of  area  by  the  radius  vector  is 

consequently 

y  r- 

Lm     irx(r*-Ar)        Lm     i         Ar_i. 
A*=02          A*          ~A*=0 2r < A t~* 

Let  f  and  r0  be  two  values  of  the  velocity  at  two  points 
P  and  P0  which  are  near  together.  The  acceleration  r0  at  P0 
is  the  limit  of 

r  —  r'0 _  Af 

A*         A  t  ' 

Break  up  the  vector  ~  =  Lz±«  into  two  components   one 
parallel  and  the  other  perpendicular  to  the  acceleration  f  0 . 
Af 


if  n  be  a  normal  to  the  vector  i?0.  The  quantity  x  ap- 
proaches  unity  when  A  t  approaches  zero.  The  quantity  y 
approaches  zero  when  A  t  approaches  zero. 

Hence     r  x  (r  -  r'0)  =  x  A*  r  x  r'0  +  y  A*  r  x  n. 
r  x  (r  -  r0)  =  r  x  i  -  (r0  +  ^  A  A  x  f0 . 


THE  DIFFERENTIAL   CALCULUS  OF  VECTORS     131 

Hence 

Ar 

rxf—  r0xi0=—  X 
A  t 

But  each  of  the  three  terms  upon  the  right-hand  side  is  an 
infinitesimal  of  the  second  order.  Hence  the  rates  of  descrip- 
tion of  area  at  P  and  P0  differ  by  an  infinitesimal  of  the 
second  order  with  respect  to  the  time.  This  is  true  for  any 
point  of  the  curve.  Hence  the  rates  must  be  exactly  equal 
at  all  points.  This  proves  the  theorem. 

60.]  The  motion  of  a  rigid  body  one  point  of  which  is 
fixed  is  at  any  instant  a  rotation  about  an  instantaneous  axis 
passing  through  the  fixed  point. 

Let  i,  j,  k  be  three  axes  fixed  in  the  body  but  moving  in 
space.  Let  the  radius  vector  r  be  drawn  from  the  fixed  point 
to  any  point  of  the  body.  Then 


But  d  r  =  (d  r  •  i)  i  +  (d  r  •  j)  j  +  (d  r  •  k)  k. 

Substituting  the  values  of  c£  r  •  i,  c?  r  •  j,  d  r  •  k  obtained  from 
the  second  equation 

dr  =  (xi  •  di+  yi  »dj  +  2i»^k)i 
+  (#  j  •  di  +  yj  •  rf  j  +  z  j  •  d  k)  j 
+  (#k«di  +  yk»c£j  +  «k»rfk)k. 

But  i  .  j  =j  .  k  =  k  .  i  =  0. 

Hence      i»rfj+j  •<£!  =  ()    or    j«(H  =  —  i»rfj 
j»dk  +  k«dj  =  0    or    k.dj  =  —  j.tfk 
k.di  +  i»dk  =  0    or    i  •  dk  =  —  'k  •  di. 

Moreover  i»i=j»j  =  k»k  =  l. 

Hence  i»^i  =    «d    =  k«c?k  =  0. 


132  VECTOR  ANALYSIS 

Substituting  these  values  in  the  expression  for  d  r. 

d  r  =  (2  i  •  d  k  —  y  j  •  d  i)  i  +  (#  j  •  d  i  —  2  k  •  d  j)  j 

+  (y  k  •  d  j  —  x  i  •  d  k)  k. 
This  is  a  vector  product. 

d  r  =  (Wj  i  +  i»dk  j  +  H i  k)x(>  i  +  y  j  +  z  k). 
Let  d  j  .  d  k  .  d  i 

a  =  k-^1  +  l-d7J+'-dlk- 

Then  .       d  r 

r  =  di=axr- 

This  shows  that  the  instantaneous  motion  of  the  body  is  one 
of  rotation  with  the  angular  velocity  a  about  the  line  a. 
This  angular  velocity  changes  from  instant  to  instant.  The 
proof  of  this  theorem  fills  the  lacuna  in  the  work  in  Art.  51. 

Two  infinitesimal  rotations  may  be  added  like  vectors. 
Let  aj  and  a2  be  two  angular  velocities.  The  displacements 
due  to  them  are 

dj  r  =  aj  x  r  d  t, 

dz  r  =  aa  x  r  d  t. 
If  r  be  displaced  by  a,  it  becomes 

T  +  d1r  =  T  +  a.1xidt. 
If  it  then  be  displaced  by  aa,  it  becomes 

r  +  d  r  =  r  +  d1  r  +  %  x  [r  +  (&l  X  r)  d  f\  d  t. 
Hence    d  r  =  ftl  x  r  d  t  +  aa  x  r  d  t  +  a2  x  (ax  x  r)  (d  t)\ 

If  the  infinitesimals  (d  t)  2  of  order  higher  than  the  first  be 
neglected, 

d  r  =  &1  x  r  d  t  +  a2  x  r  d  t  =  (&1  +  a2)  x  r  d  t, 
which  proves  the  theorem.    If  both  sides  be  divided  by  d  t 

.      dr 

r  =       =   a1  +  aa)xr. 


THE  DIFFERENTIAL   CALCULUS  OF   VECTORS      133 

This    is    the  parallelogram   law   for  angular  velocities.     It 
was  obtained  before  (Art.  51)  in  a  different  way. 

In  case  the  direction  of  a,  the  instantaneous  axis,  is  con- 
stant, the  motion  reduces  to  one  of  steady  rotation  about  a. 


r  =  a  x  r. 

The  acceleration  r=axr+axr=axr+ax  (axr). 

As  a  does  not  change  its  direction  a  must  be  collinear  with 
a  and  hence  a  x  r  is  parallel  to  a  x  r.  That  is,  it  is  perpen- 
dicular to  r.  On  the  other  hand  ax  (a  X  r)  is  parallel  to  r. 
Inasmuch  as  all  points  of  the  rotating  body  move  in  con- 
centric circles  about  a  in  planes  perpendicular  to  a,  it  is 
unnecessary  to  consider  more  than  one  such  plane. 

The  part  of  the  acceleration  of  a  particle  toward  the  centre 
of  the  circle  in  which  it  moves  is 

a  x  (a  X  r). 

This  is  equal  in  magnitude  to  the  square  of  the  angular 
velocity  multiplied  by  the  radius  of  the  circle.  It  does  not 
depend  upon  the  angular  acceleration  a  at  all.  It  corresponds 
to  what  is  known  as  centrifugal  force.  On  the  other  hand 
the  acceleration  normal  to  the  radius  of  the  circle  is 

axr. 

This  is  equal  in  magnitude  to  the  rate  of  change  of  angular 
velocity  multiplied  by  the  radius  of  the  circle.  It  does  not 
depend  in  any  way  upon  the  angular  velocity  itself  but  only 
upon  its  rate  of  change. 

61.]  The  subject  of  integration  of  vector  equations  in  which 
the  differentials  depend  upon  scalar  variables  needs  but  a 
word.  It  is  precisely  like  integration  in  ordinary  calculus. 

If  then  d  r  =  d  s, 

r  =  s  +  C, 


134  VECTOR  ANALYSIS 

where  C  is  some  constant  vector.  To  accomplish  the  integra- 
tion in  any  particular  case  may  be  a  matter  of  some  difficulty 
just  as  it  is  in  the  case  of  ordinary  integration  of  scalars. 

Example  1:  Integrate  the  equation  of  motion  of  a 
projectile. 

The  equation  of  motion  is  simply 

r=  g, 

which  expresses  the  fact  that  the  acceleration  is  always  ver- 
tically downward  and  due  to  gravity. 

r  =  g  t  +  b, 

where  b  is  a  constant  of  integration.  It  is  evidently  the 
velocity  at  the  time  t  =  0. 


c  is  another  constant  of  integration.  It  is  the  position  vector 
of  the  point  at  time  t  =  0.  The  path  which  is  given  by  this 
last  equation  is  a  parabola.  That  this  is  so  may  be  seen  by 
expressing  it  in  terms  of  x  and  y  and  eliminating  t. 

Example  2 :   The  rate  of  description  of  areas  when  a  par- 
ticle moves  under  a  central  acceleration  is  constant. 

r  =  f(r). 

Since  the  acceleration  is  parallel  to  the  radius, 
r  x  r  =  0. 

But  r  x  r  =  — -  (r  x  r). 

For  —  (r  x  f)  =  r  x  r  +  r  X  r. 

Hence  —  (r  x  j)  =  0 

cl  t 

and  r  x  r  =  C, 

which  proves  the  statement. 


THE  DIFFERENTIAL  CALCULUS  OF  VECTORS  135 

Example  3  :  Integrate  the  equation  of  motion  for  a  particle 
moving  with  an  acceleration  toward  the  centre  and  equal  to 
a  constant  multiple  of  the  inverse  square  of  the  distance 
from  the  centre. 

c2 

Given  r  =  --  3  r. 

r3 

Then  r  x  r  =  0. 

Hence  r  x  r  =  C. 

Multiply  the  equations  together  with  X. 

r  x  C     —  1  —  1       .  . 

-^2-  =  7Frx(rxr)=:  7F  {r'rr  -'•"}. 

r  •  r  =  r2. 
Differentiate.    Then        r  •  r  =  r  f  . 


Hence  121?  =  £  .  ,  £  r. 

c2  r        r2 

Each  side  of  this  equality  is  a  perfect  differential. 


Integrate.     Then         r  X  °  =  £  +  e  I, 

c2          r 

where  e  I  is  the  vector  constant  of  integration,  e  is  its  magni- 
tude and  I  a  unit  vector  in  its  direction.  Multiply  the  equa- 
tion by  r  •  . 

r  .  r  x  C      r  •  r 


+  e  r  •  I. 

C  T 

But 


C*  T 

r  »  r  x  C      rxr.CC«C 


136  VECTOR  ANALYSIS 

C  •  C 

Let  p  =  and  cos  u  =  cos  (r,  I). 

Then  p  =  r  +  e  r  cos  u. 

Or  f=  P 


6  COS  U 


This  is  the  equation  of  the  ellipse  of  which  e  is  the  eccentri- 
city. The  vector  I  is  drawn  in  the  direction  of  the  major 
axis.  The  length  of  this  axis  is 

—  drr 

1  —  e2 

It  is  possible  to  carry  the  integration  further  and  obtain 
the  time.  So  far  merely  the  path  has  been  found. 

Scalar  Functions  of  Position  in  Space.     The  Operator  V 

62.]  A  function  V  (a?,  y,  2)  which  takes  on  a  definite  scalar 
value  for  each  set  of  coordinates  #,  y,  z  in  space  is  called  a 
scalar  function  of  position  in  space.  Such  a  function,  for  ex- 

ample, is 

V  (a?,  y,  z)  =x2  +  y2  +  zz  =  r2. 

This  function  gives  the  square  of  the  distance  of  the  point 
(»,  y,  z)  from  the  origin.  The  function  V  will  be  supposed  to 
be  in  general  continuous  and  single-  valued.  In  physics  scalar 
functions  of  position  are  of  constant  occurrence.  In  the 
theory  of  heat  the  temperature  T  at  any  point  of  a  body  is  a 
scalar  function  of  the  position  of  that  point.  In  mechanics 
and  theories  of  attraction  the  potential  is  the  all-important 
function.  This,  too,  is  a  scalar  function  of  position. 

If  a  scalar  function  V  be  set  equal  to  a  constant,  the  equa- 
tion 

r(x,y,z)=c.  (20) 

defines  a  surface  in  space  such  that  at  every  point  of  it  the 
function  V  has  the  same  value  c.  In  case  F  be  the  tempera- 


THE  DIFFERENTIAL   CALCULUS  OF  VECTORS      137 

ture,  this  is  a  surface  of  constant  temperature.  It  is  called  an 
isothermal  surface.  In  case  V  be  the  potential,  this  surface  of 
constant  potential  is  known  as  an  equipotential  surface.  As 
the  potential  is  a  typical  scalar  function  of  position  in  space, 
and  as  it  is  perhaps  the  most  important  of  all  such  functions 
owing  to  its  manifold  applications,  the  surface 

V  (x,  y,z}=c 

obtained  by  setting  V  equal  to  a  constant  is  frequently  spoken 
of  as  an  equipotential  surface  even  in  the  case  where  V  has 
no  connection  with  the  potential,  but  is  any  scalar  function 
of  positions  in  space. 

The  rate  at  which  the  function  V  increases  in  the  X  direc- 
tion —  that  is,  when  x  changes  to  x  +  A  x  and  y  and  z  remain 
constant  —  is 

LIM       T  (a;  +  A  a,  y,  g)  -  T  (x,  y,  z) 


"I 
J* 


This  is  the  partial  derivative  of  Fwith  respect  to  x.  Hence 
the  rates  at  which  V  increases  in  the  directions  of  the  three 
axes  X}  F,  Z  are  respectively 

9V      9V_     9V 
9  x  '    9  y  '    Q  z 

Inasmuch  as  these  are  rates  in  a  certain  direction,  they  may 
be  written  appropriately  as  vectors.  Let  i,  j,  k  be  a  system 
of  unit  vectors  coincident  with  the  rectangular  system  of 
axes  X,  F,  Z.  The  rates  of  increase  of  V  are 

3V     .9V      u  9V 

1  -  ,    j   -  ,      K     -  . 

.    9x        9  y         9  z 

The  sum  of  these  three  vectors  would  therefore  appear  to  be 
a  vector  which  represents  both  in  magnitude  and  direction 
the  resultant  or  most  rapid  rate  of  increase  of  V.  That  this 
is  actually  the  case  will  be  shown  later  (Art.  64). 


138  VECTOR  ANALYSIS 

63.]     The  vector  sum  which  is  the  resultant  rate  of  increase 
of  F  is  denoted  by  VF. 


V  F  represents  a  directed  rate  of  change  of  V  —  a  directed 
or  vector  derivative  of  V,  so  to  speak.  For  this  reason  V  V 
will  be  called  the  derivative  of  V;  and  F,  the  primitive  of 
VF.  The  terms  gradient  and  slope  of  F  are  also  used  for 
V  F.  It  is  customary  to  regard  V  as  an  operator  which  obtains 
a  vector  VFfrom  a  scalar  function  Fof  position  in  space. 


This  symbolic  operator  V  was  introduced  by  Sir  W.  R. 
Hamilton  and  is  now  in  universal  employment.  There 
seems,  however,  to  be  no  universally  recognized  name  1  for  it, 
although  owing  to  the  frequent  occurrence  of  the  symbol 
some  name  is  a  practical  necessity.  It  has  been  found  by 
experience  that  the  monosyllable  del  is  so  short  and  easy  to 
pronounce  that  even  in  complicated  formulae  in  which  V  occurs 
a  number  of  times  no  inconvenience  to  the  speaker  or  hearer 
arises  from  the  repetition.  V  F  is  read  simply  as  "  del  F." 

Although  this  operator  V  has  been  defined  as 

.9      .  d        d 

V  =  i  —  +  i  --  h  k  —  , 

dx  3y  Qz 

1  Some  use  the  term  Nabla  owing  to  its  fancied  resemblance  to  an  Assyrian 
harp.  Others  have  noted  its  likeness  to  an  inverted  A  and  have  consequently 
coined  the  none  too  euphonious  name  Atled  by  inverting  the  order  of  the  letters  in 
the  word  Delta.  Fb'ppl  in  his  Einfuhrung  in  die  MaxweWsche  Theorie  der  Elec- 
tricitat  avoids  any  special  designation  and  refers  to  the  symbol  as  "die  Operation 
V."  How  this  is  to  be  read  is  not  divulged.  Indeed,  for  printing  no  particular 
name  is  necessary,  but  for  lecturing  and  purposes  of  instruction  something  is  re- 
quired —  something  too  that  does  not  confuse  the  speaker  or  hearer  even  when 
often  repeated. 


THE  DIFFERENTIAL   CALCULUS  OF  VECTORS     139 


so  that  it  appears  to  depend  upon  the  choice  of  the  axes,  it 
is  in  reality  independent  of  them.  This  would  be  surmised 
from  the  interpretation  of  V  as  the  magnitude  and  direction 
of  the  most  rapid  increase  of  V.  To  demonstrate  the  inde- 
pendence take  another  set  of  axes,  i',  j',  k'  and  a  new  set  of 
variables  a/,  y',  z'  referred  to  them.  Then  V  referred  to  this 
system  is 

*       +  *  <22>' 


By  making  use  of  the  formulae  (47)'  and  (47)",  Art.  53,  page 
104,  for  transformation  of  axes  from  i,  j,  k  to  i',  j',  k'  and  by 
actually  carrying  out  the  differentiations  and  finally  by 
taking  into  account  the  identities  (49)  and  (50),  V  may 
actually  be  transformed  into  V. 

V'  =  V. 

The  details  of  the  proof  are  omitted  here,  because  another 
shorter  method  of  demonstration  is  to  be  given. 
64.  ]     Consider  two  surfaces  (Fig.  30) 

V  O,  y,z)=c 
and  V  (#,  y,  z)  =  c  +  A  c, 

upon  which  Vis  constant  and  which  are  moreover  infinitely 

near  together.     Let  #,  y,  z  be  a  given  point  upon  the  surface 

V=  c.    Let  r  denote  the  ra- 

dius   vector    drawn   to    this 

point  from  any  fixed  origin. 

Then  any  point  near  by  in 

the    neighboring    surface    V 

=  c  +  d  c  may  be  represented 

by  the  radius  vector  r  +  d  r. 

The  actual  increase  of  Ffrom 

the  first  surface  to  the  second 

is  a  fixed  quantity  dc.    Tlie  rate  of  increase  is  a  variable 


FIG.  30. 


140  VECTOR  ANALYSIS 

quantity  and  depends  upon  the  direction  dr  which  is  fol- 
lowed when  passing  from  one  surface  to  the  other.  The  rate 
of  increase  will  be^the^guptieir^of  the  actual  increase  d  c  and 
tnV~fTifitapOft  Vdr  •  dr  between  the  surfaces  at  the  point 
as,  y,  2  in  the  direction  d  r.  Let  n  be  a  unit  normal  to  the 
surfaces  and  d  n  the  segment  of  that  normal  intercepted 
between  the  surfaces,  n  d  n  will  then  be  the  least  value  for 

dr.    The  quotient  , 

d  c 


yd  r • d r 

will  therefore  be  a  maximum  when  d  r  is  parallel  to  n  and 
equal  in  magnitude  of  d  n.  The  expression 

^  n  (23) 

an 

is  therefore  a  vector  of  which  the  direction  is  the  direction  of 
most  rapid  increase  of  Fand  of  which  the  magnitude  is  the 
rate  of  that  increase.  This  vector  is  entirely  independent  of 
the  axes  X,  Y,  Z.  Let  d  c  be  replaced  by  its  equal  d  V  which 
is  the  increment  of  Fin  passing  from  the  first  surface  to  the 
second.  Then  let  V  F"be  defined  again  as 


From  this  definition,  V  V  is  certainly  the  vector  which 
gives  the  direction  of  most  rapid  increase  of  V  and  the  rate 
in  that  direction.  Moreover  VFis  independent  of  the  axes. 
It  remains  to  show  that  this  definition  is  equivalent  to  the  one 
first  given.  To  do  this  multiply  by  •  d  r. 


.rfr.  (25) 

d  n 

n  is  a  unit  normal.  Hence  n  •  d  r  is  the  projection  of  d  r  on 
n  and  must  be  equal  to  the  perpendicular  distance  d  n  between 
the  surfaces. 


THE  DIFFL    ENTIAL   CALCULUS  OF  VECTORS        141 

dV 

VF.  dr  =  —  dn  =  dV.  (25)' 

dn 

9V  9V  9V 

But  dV=^-  dx+  -^-dy  +  -~-dz, 

d  x  ay  a  z 

where  (d  #)2  +  (d  y)2  +  (d  z)2  =  d  r  •  d  r. 

If  dr  takes  on  successively  the  values  i  dx,  $  dy,  bdz  the 
equation  (25)'  takes  on  the  values 

9V 
VF«  i  d  x  =  TT—  d  x 

dx 

9V 
VV.)dy  =  -rdy  (26) 

9V 
W*ls.dz  =  ^—dz. 

9z 

If  the  factors  d  x,  d  yy  d  z  be  cancelled  these  equations  state 
that  the  components  W  •  i,  VF»  j,  VF»  k  of  VT  in  the 
i,  j,  k  directions  respectively  are  equal  to 

9V_  9V  9V 

9  x*  3y*  9z' 

VF=  (VF.  i)  i  +  (VF-  j)  j  +  (VF.  k)  k. 


Hence  by(26)     VF=i        +  j+k—  .  (21) 

9  x        9  y          9z 

The  second  definition  (24)  has  been  reduced  to  the  first 
and  consequently  is  equivalent  to  it. 

*65.]  The  equation  (25V  found  above  is  often  taken  as  a 
definition  of  V  F.  Accordmg  to  ordinary  calculus  the  deriv- 

d  11 

ative  -^-  satisfies  the  equation 
d  x 

dy 


142  VECTOR  ANALYSIS 

Moreover  this  equation  defines  dy  /  dx.     In  a  similar  manner 
it  is  possible  to  lay  down  the  following  definition. 

Definition:    The  derivative  VF  of  a   scalar  function  of 
position  in  space  shall  satisfy  the  equation 


for  all  values  of  dr. 

This  definition  is  certainly  the  most  natural  and  important 
from  theoretical  considerations.  But  for  practical  purposes 
either  of  the  definitions  before  given  seems  to  be  better. 
They  are  more  tangible.  The  real  significance  of  this  last 
definition  cannot  be  appreciated  until  the  subject  of  linear 
vector  functions  has  been  treated.  See  Chapter  VII. 

The  computation  of  the  derivative  V  of  a  function  is  most 
frequently  carried  on  by  means  of  the  ordinary  partial 
differentiation. 


Example  1 :    Let     V  (x,  y,z)—r  =  V 

.  dr      .dr         3r 
i^-+J^r+k^- 
ax         dy         &  z 


Vr=:i 


.    ,  , 

Vx*  +  yz  +  z2        Vx*  +  y'*  +  z* 

z 


_ 
Vx*  +  y'*  +  zz 

Hence      v  r  =  i* 


and  V  r  =      *          * 

r 


The  derivative  of  r  is  a  unit  vector  in  the  direction  of  r. 
This  is  evidently  the  direction  of  most  rapid  increase  of  r 
and  the  rate  of  that  increase. 


THE  DIFFERENTIAL   CALCULUS  OF  VECTORS    143 
:    Let  1  1 


r      y  x2  -f  y2  +  22 

_i       y 

(xz  +  v2  +  z2)' 

2 


<>2  +  2/2  +  22)1 

Hence      V  -  =  — 5—  -=- — ^-  (-  i  x  -  j  y  -  k  2) 
r      (a;2  +  y2  +  22) » 

^_         r         j 

and  V  -  = -T  =  -Y  • 

r      (r  •  r)?       r3 

The  derivative  of  1/r  is  a  vector  whose  direction  is  that 
of  —  r,  and  whose  magnitude  is  equal  to  the  reciprocal  of  the 
square  of  the  length  r. 


Example  3 :  V  r1*  =  n  r  *    1  =  711* • 

r  •  r 

The  proof  is  left  to  the  reader. 
Example  4  '  Let         F(#,  y,  2)  =  log  yx2  -f  v3. 

a;  y 

(i «  +  j  y). 


If  r  denote  the  vector  drawn  from  the  origin  to  the  point 
(#,  y,  2)  of  space,  the  function  V  may  be  written  as 

F(#,y, 2)  =  log  Vr  •  r  —  (k  •  r)2 
and  ia?  +  jy  =  r  —  k  k»r. 

/ — u &          r  —  k  k«r 

Hence  V log  V #2  +  #2  = „    .„ 

r  •  r  —  ^K»rj* 

r  —  k  k»r 
=  (r-kk.r).(r-kk.r)' 


144  VECTOR  ANALYSIS 

There  is  another  method  of  computing  V  which  is  based 
upon  the  identity 


Example  1  :  Let  F  =  Vr«r  =  r. 


YT  •  r  V  r  •  r 

Hence  ^7^=-^=  =  -' 

V  r«r      r 

Example  2  :    Let     F  =  r  •  a,  where  a  is  a  constant  vector. 

dF=rfr-a  =  <2r.VF. 
Hence  VF=a. 

Example  3:    Let  F=  (rxa)  •  (rxb),  where  a  and  b  are 

constant  vectors. 

F  =  r«r  a»b  —  r»a  r«b.  , 

Ar.rtub*     i-.*-  «.  la  >  r<<  '   *T« 

d  F  =  2  dr»r  a»b  -  dr-a  r-b  -  dr«b  r«a  =  <f  r  •  V  F. 


Hence  VF=  2ra«b  —  ar.b  —  br«a 

VF=  (ra.b-ar.b)  +  (ra»b-br.a) 
=  bx(rxa)  +  a  x  (rxb). 

Which  of  these  two  methods  for  computing  V  shall  be 
applied  in  a  particular  case  depends  entirely  upon  their 
relative  ease  of  execution  in  that  case.  The  latter  method  is 
independent  of  the  cob'rdinate  axes  and  may  therefore  be 
preferred.  It  is  also  shorter  in  case  the  function  Fcan  be 
expressed  easily  in  terms  of  r.  But  when  F  cannot  be  so 
expressed  the  former  method  has  to  be  resorted  to. 

*66.]  The  great  importance  of  the  operator  V  in  mathe- 
matical physics  may  be  seen  from  a  few  illustrations.  Sup- 
pose T  (#,  y,  2)  be  the  temperature  at  the  point  #,  y,  2  of  a 


THE  DIFFERENTIAL  CALCULUS   OF  VECTORS     145 

heated  body.  That  direction  in  which  the  temperature  de- 
creases most  rapidly  gives  the  direction  of  the  flow  of  heat. 
V  jP,  as  has  been  seen,  gives  the  direction  of  most  rapid 
increase  of  temperature.  Hence  the  flow  of  heat  f  is 

f  =  -k  Vr, 

where  Jc  is  a  constant  depending  upon  the  material  of  the 
body.  Suppose  again  that  V  be  the  gravitational  potential 
due  to  a  fixed  body.  The  force  acting  upon  a  unit  mass  at 
the  point  (#,  y,  z)  is  in  the  direction  of  most  rapid  increase  of 
potential  and  is  in  magnitude  equal  to  the  rate  of  increase 
per  unit  length  in  that  direction.  Let  F  be  the  force  per  unit 

mass.     Then 

F  =  VF. 

As  different  writers  use  different  conventions  as  regards  the 
sign  of  the  gravitational  potential,  it  might  be  well  to  state 
that  the  potential  V  referred  to  here  has  the  opposite  sign  to 
the  potential  energy.  If  W  denoted  the  potential  energy  of 
a  mass  m  situated  at  #,  £/,  2,  the  force  acting  upon  that  mass 
would  be 


In  case  V  represent  the  electric  or  magnetic  potential  due 
to  a  definite  electric  charge  or  to  a  definite  magnetic  pole  re- 
spectively the  force  F  acting  upon  a  unit  charge  or  unit  pole 

as  the  case  might  be  is 

F  =  -  VF. 

The  force  is  in  the  direction  of  most  rapid  decrease  of 
potential.  In  dealing  with  electricity  and  magnetism  poten- 
tial and  potential  energy  have  the  same  sign  ;  whereas  in 
attraction  problems  they  are  generally  considered  to  have 
opposite  signs.  The  direction  of  the  force  in  either  case  is  in 
the  direction  of  most  rapid  decrease  of  potential  energy.  The 
difference  between  potential  and  potential  energy  is  this. 

10 


146  VECTOR  ANALYSIS 

Potential  in  electricity  or  magnetism  is  the  potential  energy 
per  unit  charge  or  pole ;  and  potential  in  attraction  problems 
is  potential  energy  per  unit  mass  taken,  however,  with  the 
negative  sign. 

*67.]  It  is  often  convenient  to  treat  an  operator  as  a 
quantity  provided  it  obeys  the  same  formal  laws  as  that 
quantity.  Consider  for  example  the  partial  differentiators 

JL    _?__?_. 

9x*  9y*  9z 

As  far  as  combinations  of  these  are  concerned,  the  formal  laws 
are  precisely  what  they  would  be  if  instead  of  differentiators 

three  true  scalars 

a,     &,    c 

were  given.     For  instance 
the  commutative  law 


d     9        99 

—  —  =  —  -r—  ~  a  o  =  o  a, 
9x9y     9y9x 


the  associative  law 


9/99\/99\9  ~  v      /  w 

;r-  ( ;r-  ;r~  1  =  (  ir~  TT~  )  « a  (6  c)  =  (a  6)c, 

9z\9y9zJ      \9x9yj9z 


and  the  distributive  law 


9/9        9  \       99        99 

-  }  =  ---  ----  a 

9x\dy     5zJ     9x9y     9x9z 


9 

hold  for  the  differentiators  just  as  for  scalars.    Of  course  such 

formulae  as 

9        9 

w^  =  aiM> 

where  u  is  a  function  of  x  cannot  hold  on  account  of  the 
properties  of  differentiators.  A  scalar  function  u  cannot  be 
placed  under  the  influence  of  the  sign  of  differentiators. 
Such  a  patent  error  may  be  avoided  by  remembering  that  an 
operand  must  be  understood  upon  which  9/9  #  is  to  operate. 


THE  DIFFERENTIAL   CALCULUS   OF  VECTORS    147 

In  the  same  way  a  great  advantage  may  be  obtained  by 

looking  upon 

Sf\  f\ 

V  —  4-    *  —  4-  k  — 

9  x          dy          dz 
as  a  vector.    It  is  not  a  true  vector,  for  the  coefficients 


are  not  true  scalars.  It  is  a  vector  differentiator  and  of 
course  an  operand  is  always  implied  with  it  As  far  as  formal 
operations  are  concerned  it  behaves  like  a  vector.  For 
instance 


V  (u  0)  =  (V  u)  i)  +  u  (V  0), 
c  V  u  =  V  (c  M), 

if  u  and  v  are  any  two  scalar  functions  of  the  scalar  variables 
#,  y,  z  and  if  c  be  a  scalar  independent  of  the  variables  with 
regard  to  which  the  differentiations  are  performed. 

68.]  If  A  represent  any  vector  the    formal  combination 
A.  Vis 

A-v=^+^ii;  +  ^a4-'       <2T> 

provided  A  =  Al  i  +  -42  j  +  A3k. 

This  operator  A  •  V  is  a  scalar  differentiator.  When  applied 
to  a  scalar  function  V  (x,  y,  z)  it  gives  a  scalar. 

0  V  e$  V  0  V 

^r=Al-+A,-+A3-.        (28) 


Suppose  for  convenience  that  A  is  a  unit  vector  a. 

C/  vC  { 


-+a3-  (29) 


148  VECTOR  ANALYSIS 

where  av  «2,  «3  are  the  direction  cosines  of  the  line  a  referred 
to  the  axes  X,  Y,  Z.     Consequently  (a  •  V)  F  appears  as  the 
well-known  directional   derivative  of   F  in  the  direction   a. 
This  is  often  written 


h  a 

5s          3x          3y  z 

It  expresses  the  magnitude  of  the  rate  of  increase  of  F  in 
the  direction  a.  In  the  particular  case  where  -this  direction  is 
the  normal  n  to  a  surface  of  constant  value  of  F,  this  relation 
becomes  the  normal  derivative. 


if  nv  nv  «3  be  the  direction  cosines  of  the  normal. 

The  operator  a  •  V  applied  to  a  scalar  function  of  position 
F  yields  the  same  result  as  the  direct  product  of  a  and  the 

vector  V  F. 

(a.V)F=a.(VF).  (30) 

For  this  reason  either  operation  may  be  denoted  simply  by 

a.VF 

without  parentheses  and  no  ambiguity  can  result  from  the 
omission.  The  two  different  forms  (a  •  V)  Fand  a  •  (V  F) 
may  however  be  interpreted  in  an  important  theorem. 
(a  .  V)  F  is  the  directional  derivative  of  F  in  the  direction 
a.  On  the  other  hand  a  •  (V  F)  is  the  component  of  V  F  in 
the  direction  a.  Hence  :  The  directional  derivative  of  F  in 
any  direction  is  equal  to  the  component  of  the  derivative 
VFin  that  direction.  If  Fdenote  gravitational  potential  the 
theorem  becomes  :  The  directional  derivative  of  the  potential 
in  any  direction  gives  the  component  of  the  force  per  unit 
mass  in  that  direction.  In  case  Fbe  electric  or  magnetic 
potential  a  difference  of  sign  must  be  observed. 


THE  DIFFERENTIAL   CALCULUS  OF  VECTORS    149 

Vector  Functions  of  Position  in  Space 
69.]     A  vector  function  of  position  in  space  is  a  function 


-  \  *  ^i^r  *«-—  v  (x  v  z\ 

&  v:  •5>MJ      "*?•  v    '  yj    ' 

which  associates  with  each  point  x,  y,  z  in  space  a  definite 
vector.  The  function  may  be  broken  up  into  its  three  com- 
ponents 

V  (x,  y,  z)  =  F!  (x,  y,z)i+  F2  (as,  y,  z)  j  +  F3  (x,  y,  z)  k. 

Examples  of  vector  functions  are  very  numerous  in  physics. 
Already  the  function  VF  has  occurred.  At  each  point  of 
space  V  F  has  in  general  a  definite  vector  value.  In  mechan- 
ics of  rigid  bodies  the  velocity  of  each  point  of  the  body  is  a 
vector  function  of  the  position  of  the  point.  Fluxes  of  heat, 
electricity,  magnetic  force,  fluids,  etc.,  are  all  vector  functions 
of  position  in  space. 

The  scalar  operator  a  •  V  may  be  applied  to  a  vector  func- 
tion V  to  yield  another  vector  function. 

Let     V  =  Vl  (x,  y,  z),  i  +  F2  (a,  y,z)j+  F3  (as,  y,  z)  k 
and  a  =  al  i  +  «2  j  +  a3  k. 

9  9  5 

Then  a  .  V  =  04 h  «2  - — l~a3T— 

Q  x  3  y  o  z 

(a.V)V  =  (a-V)  Vl  i  +  (a .  V)  F2j   +(a.V)F3  k 

9Vl 


and    (a.V)V 

(31) 


150  VECTOR  ANALYSIS 

This  may  be  written  in  the  f  oral 


Hence  (a  •  V)  V  is  the  directional  derivative  of  the  vector 
function  V  in  the  direction  a.  It  is  possible  to  write 

(a  .  V)  V  =  a  •  VV 

without  parentheses.  For  the  meaning  of  the  vector  symbol 
V  when  appliod  to  ft-  yp.pt.nr  fn  notion  V  has  not  yet  been 
dctiiu'd.  Hence  from  the  present  standpoint  the  expression 
a  •  VV  can  have  but  the  one  interpretation  given  to  it  by 
(a.V)  V. 

70.]  Although  the  operation  V  V  has  not  been  defined  and 
cannot  be  at  present,1  two  formal  combinations  of  the  vector 
operator  V  and  a  vector  function  V  may  be  treated.  These 
are  the  (formal)  scalar  product  and  the  (formal)  vector  prod- 
uct of  V  into  V.  They  are 

T     <32> 

and  VxV  =    i       +  j        +  k_xV.  (33) 

V  •  V  is  read  del  dot  V;  and  V  x  V,  del  cross  V. 

500 
The  differentiators  —  -  ,  -—  ,  —  ,  being  scalar  operators,  pass 

by  the  dot  and  the  cross.     That  is 


=lx+Jx        +  kx.         (33)' 

These  may  be  expressed  in  terms  of  the  components  Vv  Vv  Vz 
of  V. 

1  A  definition  of  A  V  will  be  given  in  Chapter  VTL 


Then 


Hence 
Moreover 


Hence 


Now.        t  -    ^•  =  ^-i 


T^^  DIFFERENTIAL   CALCULUS  OF  VECTORS     151 

PV_3F1  .  .  3F2.  ,  3F3v 

(34) 


9x      9x          9x         9x 
3V     3F,        3F9  .      3F- 

_ i  i     i    «  i    i     2  V 

3  y         9  y 

2  i    i     _      8  ^ 

J  -          —  »• 


3V 
9z 


3V_3Fj 

Q)  tC  c)  ZC 


3V 


v.v  =  ^  +  ^  +  ^^ 


3V      .  3F3 
j  X  —  =  i  -r-2  -  k 


3V=    3FJ_. 

3s          3s  3s 


* 


\  3a;        3y  J 
This  may  be  written  in  the  form  of  a  determinant 


Vx  V  = 


i        j         k 

999 
9 x     9  y     3s 


K*j- 

C*K 


(33)'" 


152  VECTOR  ANALYSIS 

It  is  to  be  understood  that  the  operators  are  to  be  applied  to 
the  functions  Vv  Vv  Vz  when  expanding  the  determinant. 

From  some  standpoints  objections  may  be  brought  forward 
against  treating  V  as  a  symbolic  vector  and  introducing  V  •  V 
and  V  x  V  respectively  as  the  symbolic  scalar  and  vector 
products  of  V  into  V.  These  objections  may  be  avoided  by 
simply  laying  down  the  definition  that  the  symbols  V  •  and 
V  x,  which  may  be  looked  upon  as  entirely  new  operators 
quite  distinct  from  V,  shall  be 


=  ix+jx+kx.       (33)' 


But  for  practical  purposes  and  for  remembering  formulae  it 
seems  by  all  means  advisable  to  regard 


9x  "  Qy          9z 

as  a  symbolic  vector  differentiator.     This  symbol  obeys  the 
same  laws  as  a  vector  just  in  so  far  as  the  differentiators 

^ — »  •«— » -«—  obey  the  same  laws  as  ordinary  scalar  quantities. 

71.]  That  the  two  functions  V  •  V  and  V  x  V  have  very 
important  physical  meanings  in  connection  with  the  vector 
function  V  may  be  easily  recognized.  By  the  straight- 
forward proof  indicated  in  Art.  63  it  was  seen  that  the 
operator  V  is  independent  of  the  choice  of  axes.  From  this 
fact  the  inference  is  immediate  that  V  •  V  and  V  x  V  represent 
intrinsic  properties  of  V  invariant  of  choice  of  axes.  In  order 
to  perceive  these  properties  it  is  convenient  to  attribute  to  the 
function  V  some  definite  physical  meaning  such  as  flux  or 
flow  of  a  fluid  substance.  Let  therefore  the  vector  V  denote 


153 


at  each  point  of  space  the  direction  and  the  magnitude  of  the 
flow  of  some  fluid.  This  may  be  a  material  fluid  as  water 
or  gas,  or  a  fictitious  one  as  heat  or  electricity.  To  obtain  as 
great  clearness  as  possible  let-  the  fluid  be  material  but  not 
necessarily  restricted  to  incompressibility  like  water. 


Then  v.V  = 

d  x  9y  dz 

is  called  the  divergence  of  V  and  is  often  written 
V«V=div  V. 

The  reason  for  this  term  is  that  V»V  gives  at  each  point  the 
rate  per  unit  volume  per  unit  time  at  which  fluid  is  leaving 
that  point  —  the  rate  of  diminution  of  density.  To  prove 
this  consider  a  small  cube  of  matter  (Fig.  31).  Let  the  edges 
of  the  cube  be  dx,  dy,  and  dz  respectively.  Let 

V  O,  y,  z)  =  Vl  (x,  y,  z)  i  +  V^  (x,  y,  z)  j  +  Vz  (x,  yy  z)  k. 

Consider  the  amount  of  fluid  which  passes  through   those 

faces  of  the  cube  which  are  parallel  to  the  F^-plane,  i.  e. 

perpendicular  to  the  X 

axis.     The  normal  to  the 

face  whose  x  coordinate  is 

the  lesser,  that  is,  the  nor- 

mal to  the  left-hand  face 

of  the  cube  is  —  i.  The  flux 

of  substance  through  this 

face  is 

—  i  •  V  (x,  y,  z)  dy  dz. 

The  normal  to  the  oppo-  z  F      31 

site   face,  the  face  whose 

x  coordinate  is  greater  by  the  amount  dx,  is  +  i  and  the  flux 

through  it  is  therefore 


-i  d.y 


154  VECTOR  ANALYSIS 

[3V       "I 
V(x,y,z)  +  —  dx\  dydz 

#V 
=  i  •  V  (x,  y,  z)  dy  dz  +  i  •  —  dx  dy  dz. 

<y  <X/ 

The  total  flux  outward  from  the  cube  through  these  two 
faces  is  therefore  the  algebraic  sum  of  these  quantities.  This 
is  simply 

i  .  —  —  dx  dy  dz  —  -^—  ^  dx  dy  dz. 
d  x  ox 

In  like  manner  the  fluxes  through  the  other  pairs  of  faces  of 
the  cube  are 

3  V  7  j  1.3V  ,     ,     , 

i  .  -  —  dx  dy  dz  and  k  •  •=  —  dx  dy  dz. 

&  y  &  Z 

The  total  flux  out  from  the  cube  is  therefore 

3V  3V      .      3V 


/.    3V  3V 

(  i  •  T-  •  +  J  •  o~ 

\      dx  9y 


dx  d  dz- 


This  is  the  net  quantity  of  fluid  which  leaves  the  cube  per 
unit  time.  The  quotient  of  this  by  the  volume  dx  dy  dz  of 
the  cube  gives  the  rate  of  diminution  of  density.  This  is 

3V          3V  3V      3F,      3F2      3F3 

V.V=i.    —  +  j  .  —  -  +  k  .  _  =  —  -i  +  __  ^  +  —  -*. 
c/a;  d  y  d  z        a  x         ay          az 

Because  V  •  V  thus  represents  the  diminution  of  density 
or  the  rate  at  which  matter  is  leaving  a  point  per  unit  volume 
per  unit  time,  it  is  called  the  divergence.  Maxwell  employed 
the  term  convergence  to  denote  the  rate  at  which  fluid  ap- 
proaches a  point  per  unit  volume  per  unit  time.  This  is  the 
negative  of  the  divergence.  In  case  the  fluid  is  incompressible, 
as  much  matter  must  leave  the  cube  as  enters  it.  The  total 
change  of  contents  must  therefore  be  zero.  For  this  reason 
the  characteristic  differential  equation  which  any  incompres- 
sible fluid  must  satisfy  is 


THE  DIFFERENTIAL   CALCULUS   OF   VECTORS    155 

where  V  is  the  flux  of  the  fluid.  This  equation  is  often 
known  as  the  hydrodynamic  equation.  It  is  satisfied  by  any 
flow  of  water,  since  water  is  practically  incompressible.  The 
great  importance  of  the  equation  for  work  in  electricity  is  due 
to  the  fact  that  according  to  Maxwell's  hypothesis  electric  dis- 
placement obeys  the  same  laws  as  an  incompressible  fluid.  If 
then  D  be  the  electric  displacement, 

div  D  =  V  •  D  =  0. 

72.]  To  the  operator  V  x  Maxwell  gave  the  name  curl. 
This  nomenclature  has  become  widely  accepted. 

V  x  V  =  curl  V. 

The  curl  of  a  vector  function  V  is  itself  a  vector  function 
of  position  in  space.  As  the  name  indicates,  it  is  closely 
connected  with  the  angular  velocity  or  spin  of  the  flux  at 
each  point.  But  the  interpretation  of  the  curl  is  neither  so 
easily  obtained  nor  so  simple  as  that  of  the  divergence. 

Consider  as  before  that  V  represents  the  flux  of  a  fluid. 
Take  at  a  definite  instant  an  infinitesimal  sphere  about  any 
point  (#,  y,  z).  At  the  next  instant  what  has  become  of  the 
sphere  ?  In  the  first  place  it  may  have  moved  off  as  a  whole 
in  a  certain  direction  by  an  amount  d  r.  In  other  words  it 
may  have  a  translational  velocity  of  di/dt.  In  addition  to 
this  it  may  have  undergone  such  a  deformation  that  it  is  no 
longer  a  sphere.  It  may  have  been  subjected  to  a  strain  by 
virtue  of  which  it  becomes  slightly  ellipsoidal  in  shape. 
Finally  it  may  have  been  rotated  jis_  a  whole  about  some 
axis  through  an  angle  dw.  That  is  to  say,  it  may  have  an 
angular  velocity  the  magnitude  of  which  is  dw/dt.  An 
infinitesimal  sphere  therefore  may  have  any  one  of  three 
distinct  types  of  motion  or  all  of  them  combined.  Firsjb,  a 
translation  with  definite  velocity.  Second,  a  strain  with  three 
definite  rates  of  elongation  along  the  axes  of  an  ellipsoid. 


156  VECTOR  ANALYSIS 

Th0d,  an  angular  velocity  about  a  definite  axis.  It  is  this 
third  type  of  motion  which  is  given  by  the  curl.  In  fact, 
the  curl  of  the  flux  V  is  a  vector  which  has  at  each  point  of 
space  the  direction  of  the  instantaneous  axis  of  rotation  at 
that  point  and  a  magnitude  equal  to  twice  the  instantaneous 
angular  velocity  about  that  axis. 

The  analytic  discussion  of  the  motion  of  a  fluid  presents 
more  difficulties  than  it  is  necessary  to  introduce  in  treating 
the  curl.  The  motion  of  a  rigid  body  is  sufficiently  complex 
to  give  an  adequate  idea  of  the  operation.  It  was  seen  (Art. 
51)  that  the  velocity  of  the  particles  of  a  rigid  body  at  any 
instant  is  given  by  the  formula 

v  =  v0  +  a  x  r. 

curl  v  =  Vxv  =  Vxv0  +  Vx  (axr). 
Let  a  =  a1  i  +  a2  j  +  «3  k 


expand  V  x  (a  x  r)  formally  as  if  it  were  the  vector  triple 
product  of  V,  a,  and  r.  Then 

V  x  v  =  V  x  v0  +  (V  •  r)  a  -  (V  -  a)  r. 
v0  is  a  constant  vector.     Hence  the  term  V  x  v0  vanishes. 

_dx     9y     dz_ 
v  •  r  =  ^  —  1-«  —  \-  —  =  6. 

dx       dy      dz 

As  a  is  a  constant  vector  it  may  be  placed  upon  the  other  side 
of  the  differential  operator,  V  •  a  =  a  •  V. 

(r\  «  f\    » 

«l^+a2^+"3^ 

Hence  Vxv  =  3a  —  a  =  2a. 

Therefore  in  the  case  of  the  motion  of  a  rigid  body  the  curl 
of  the  linear  velocity  at  any  point  is  equal  to  twice  the 
angular  velocity  in  magnitude  and  in  direction. 


THE  DIFFERENTIAL   CALCULUS  OF  VECTORS     157 
V  x  v  =  curl  v  =  2  a, 
a  =  gVxv  =  2  curl  v. 
v  =  v0  +  \  (V  x  v)  x  r  =  Vo  +  \  (curl  v)  x  r.      (34)  , 

The  expansion  of  V  X  (a  x  r)  formally  may  be  avoided  by 
multiplying  a  x  r  out  and  then  applying  the  operator  V  X  to 
the  result. 

73.]  It  frequently  happens,  as  in  the  case  of  the  applica- 
tion just  cited,  that  the  operators  V>V*>  V  x,  have  to  be 
applied  to  combinations  of  scalar  functions,  vector  functions, 
or  both.  The  following  rules  of  operation  will  be  found 
useful.  Let  w,  v  be  scalar  functions  and  u,  v  vector  func- 
tions of  position  in  space.  Then 


V  (u  +  v)  =  Vu  +  Vv  (35) 

V.(u  +  v)  =  V.u  +  V.v  (36) 

Vx(u  +  v)  =  Vxu  +  Vxv  (37) 

V  (u  v)  =  v  V  u  +  u  V  v  (38) 

V.(wv)  =  Vwv  +  wV«v  (39) 

Vx  («  v)  B  V  *  x  .?'+  «  V  X  ?  (40) 

'      V(u»v)  =  v.  Vn  +  u«  Vv  (41) 

+  v  x  (V  x  u)  +  n  x  (V  x  v)1 

V«(uxv)=v.Vxu  —  u.Vxv          (42) 
V  X  (u  x  v)  =  v.  Vu  —  vV-u  —  n.  Vv  +  uV.v.1    (43) 

A  word  is  necessary  upon  the  matter  of  the  interpretation 
of  such  expressions  as 

Vwv,         Vw.v,       'Vu  X  v. 

The  rule  followed  in  this  book  is  that  the  operator  V  applies 
to  the  nearest  term  only.     That  is, 

1  By  Art.  69  the  expressions  v  •  V  u  an(i  u  •  V  v  are  *°  be  interpreted  as 
«  V)v- 


158  VECTOR  ANALYSIS 

V  u  v  —  (V  u~)  v 

V  u  •  v  =  (  V  u)  •  v 

V  u  x  v  =  (  V  u)  x  v. 

If  V  is  to  be  applied  to  more  than  the  one  term  which  follows 
it,  the  terms  to  which  it  is  applied  are  enclosed  in  a  paren- 
thesis as  upon  the  left-hand  side  of  the  above  equations. 

The  proofs  of  the  formulae  may  be  given  most  naturally 
by  expanding  the  expressions  in  terms  of  three  assumed  unit 
vectors  i,  j,  k.  The  sign  2  of  summation  will  be  found  con- 
venient. By  means  of  it  the  operators  V»  V*»  A  x  take  the 
form 


The  summation  extends  over  #,  y,  z. 
To  demonstrate        Vx    wv 


2. 

Hence  Vx(wv 

To  demonstrate 

V  (u  .  v)  =  v  .  V  u  +  u  .  Vv  +  v  x  (V  x  u)  +  u  x  (V  x  v). 


THE  DIFFERENTIAL  CALCULUS  OF  VECTORS    159 

'  9  x) 


P  a 


Now 


•v*       5  u  .  -v->       .  P  u 

2v.^1  =  vx(Vxu)  +  2v..- 

or  2  v  •  ^—  i  =  v  x  (V  x  u)  +  v  •  Vu. 

-* 


^  g 

In  like  manner  ^n«  —  i  =  ux  (V  xv)  +  u»Vv. 
^      Bx 


Hence    V(n»v)  = 

+  v  x  (V  x  u)  +  u  x  (V  x  v). 

The  other  formulae  are  demonstrated  in  a  similar  manner. 

71]    The  notation1 

V(n.v)u  (44) 

will  be  used  to  denote  that  in  applying  the  operator  V  to  the 
product  (u  •  v),  the  quantity  n  is  to  be  regarded  as  constant. 
That  is,  the  operation  V  is  carried  out  only  partially  upon 
the  product  (u  •  v).  In  general  if  V  is  to  be  carried  out 
partially  upon  any  number  of  functions  which  occur  after 
it  in  a  parenthesis,  those  functions  which  are  constant  for  the 
differentiations  are  written  after  the  parenthesis  as  subscripts. 

Let  u  =  MJ  i  +  w2j  +  w3k, 


1  This  idea  and  notation  of  a  partial  V  so  to  speak  may  be  avoided  by  means 
of  the  formula  41.  But  a  certain  amount  of  compactness  and  simplicity  is 
lost  thereby.  The  idea  of  V  (u  '  v)n  's  surely  no  more  complicated  than  u  •  V  v  or 
v  X  (V  X  u). 


160  VECTOR  ANALYSIS 

then  M.V  =  UIVI  +  u2v%  +  usv3 

f\ 
and        V  (u  •  v)  =  2  i     ~  0*ivi  +  u*v*  + 


But 

and 


<:t 

Hence  V(u«  v)  =  v1  Vw}  +  v2  V  w2  + 


But  V(tt.¥)m  =  «1Vv1  +  waVi?a  +  M8V«l      (44)' 

and  V(u.v)v  =  -y1Vw1  +  v2Vw2  +  v3V^3. 

Hence  V  (u  •  v)  =  V  (u  •  v)n  +  V  (u  •  v)  v.  (45) 

This  formula  corresponds  to  the  following  one  in  the  nota- 
tion of  differentials 

d  (u  •  v)  =  d  (u  •  v)n  +  d  (u  •  v)T 
or  d  (u  •  v)  =  u  •  d  v  +  d  u  •  v. 

The  formulae  (35)-(43)  given  above  (Art.  73)  may  be 
written  in  the  following  manner,  as  is  obvious  from  analogy 
with  the  corresponding  formulae  in  differentials  : 

V  (M  +  v)  =  V  (u  +  v~)u  +  V  (u  +  «).        (35)' 

V-  (n  +  v)  =  V.  (u  +  v)u  +  V.  (u  +  v)y     (36)' 

V  x  (u  +  v)  =  V  x  (u  +  v)u  +  V  x  (u  +  v)v     (37)' 


THE  DIFFERENTIAL   CALCULUS  OF  VECTORS    161 

V  O  v)  =  V  (u  v\  +  V  (u  v).  (38)' 

V.(wv)=  V.(wv)«  +  V-(wv)v  (39)' 

V  x  (u  v)  =  V  x  (u  v)tt  +  V  x  (w  v)v  (40)' 

V  (u.  v)  =  V  (u.  v)u  +  V  (u.  v)y  (41)' 

V  •  (u  x  v)  =  V  '•  (u  x  v)u  +  V  .  (n  x  v)v  (42)' 

V  x  (u  x  v)  =  V  x  (u  x  v)u  +  V  x  (u  x  v)v.  (43)' 

This  notation  is  particularly  useful  in  the  case  of  the 
scalar  product  u«v  and  for  this  reason  it  was  introduced. 
In  almost  all  other  cases  it  can  be  done  away  without  loss  of 
simplicity.  Take  for  instance  (43)'.  Expand  V  x  (u  x  v)u 

formally. 

V  x  (u  x  v)u  =  (V  •  v)  u  —  (V  •  u)  v, 

where  it  must  be  understood  that  n  is  constant  for  the  differ- 
entiations which  occur  in  V.  Then  in  the  last  term  the 
factor  u  may  be  placed  before  the  sign  V.  Hence 


V  x  (u  x  v)u  = 
In  like  manner    V  x  (n  x  v)v  =  v  •  V  u  —  v  V  •  n. 
Hence      Vx(uxv)=v»Vu  —  v  V  •  u  —  u  •  V  v  +  u  V  •  v. 

75.]  There  are  a  number  of  important  relations  in  which 
the  partial  operation  V  (u  •  v)u  figures. 

u  x  (Vx  v)  =  V(n.v)n-u«Vv,  (46) 

or  V  (u.v)u  =  u.  Vv  +  u  x  (V  x  v),  (46)' 

or  u  •  V  v  =  V  (u  •  v)u  +  (V  x  v)  x  n.  (46)" 

The  proof  of  this  relation  may  be  given  by  expanding  in 
terms  of  i,  j,  k.  A  method  of  remembering  the  result  easily 
is  as  follows.  Expand  the  product 

u  x  (V  x  v) 
11 


162  VECTOR  ANALYSIS 

formally  as  if  V,  u,  v  were  all  real  vectors.     Then 
ux(Vxv)  =  u»vV  —  u  •  V  v. 

The  second  term  is  capable  of  interpretation  as  it  stands. 
The  first  term,  however,  is  not.  The  operator  V  has  nothing 
upon  which  to  operate.  It  therefore  must  be  transposed  so 
that  it  shall  have  u  •  v  as  an  operand.  But  u  being  outside 
of  the  parenthesis  in  u  x  (V  x  v)  is  constant  for  the  differen- 
tiations. Hence 

u .  v  V  =  V  (n  •  v)n 

and  u  x  (V  x  v)  =  V  (u .  v)u  -  u  •  V  v.  (46) 

If  u  be  a  unit  vector,  say  a,  the  formula 

a  •  V  v  =  V  (a  •  v)a  +  (V  x  v)  x  a  (47) 

expresses  the  fact  that  the  directional  derivative  a  •  V  v  of  a 
vector  function  v  in  the  direction  a  is  equal  to  the  derivative 
of  the  projection  of  the  vector  v  in  that  direction  plus  the 
vector  product  of  the  curl  of  v  into  the  direction  a. 
Consider  the  values  of  v  at  two  neighboring  points. 

v  (a,  y,  z) 
and  v  (x  +  d  #,  y  +  d  y,  z  +  d  z) 

d  v  =  v  (x  +  dx,  y  +  dy,  z  +  dz)  —  v  (»,  y,  z). 
Let  v  =  vli  +  v2  j  +  v3k 

d  v  =  d  vl  i  +  d  vz  j  +  d  VB  k. 
But  by  (25)'  dv1  =  dr»Vv1 

dvz  =  dr»  Vfl2 
dv3  =  dr»  Vv8. 

Hence         dv  =  dr.  (V vti  +  Vt>2  j  +  V v8k). 
Hence  d  v  =  d  r  •  V  ve 

By  (46)"        d  v  =  V  (d  r .  v)dr  +  (V  x  v)  x  d  r.          (48) 


THE  DIFFERENTIAL  CALCULUS  OF  VECTORS     163 

Or  if  v0  denote  the  value  of  v  at  the  point  (#,  y,  z)  and  v  the 
value  at  a  neighboring  point 

v  =  v0  +  V  (d  r  •  v)at  +  (V  x  v)  x  dr.         (49) 

This  expression  of  v  in  terms  of  its  value  v0  at  a  given  point, 
the  dels,  and  the  displacement  d  r  is  analogous  to  the  expan- 
sion of  a  scalar  functor  of  one  variable  by  Taylor's  theorem, 

/  (a)  =/(*o)  +/'(««)<*  *- 

The  derivative  of  (r  •  v)  when  v  is  constant  is  equal  to  v. 
That  is  V  (r  •  v)v  =  v. 

For  V  (r  •  v)v  =  v  •  V  r  -  (V  x  r)  x  v, 


999 

V  V  —  Vl  TT-   +  V2  T~+  VB   0~ 

1  9  x        *  9  y        d  9z 


v«  Vr  =  vli  +  v2j  +  ^3k  =  v, 

V  x  r  =  0. 
Hence  V  (r  •  v)v  =  v. 

In  like  manner  if  instead  of  the  finite  vector  r,  an  infinitesimal 
vector  d  r  be  substituted,  the  result  still  is 

V  (d  r  •  v)v  =  v. 

By  (47)        v  =  v0  +  V(dr.v)dr+  (V  x  v)  x  rfr 
V(dr-v)  =  V(rfr.v)d>  +  V^r-vV 

Hence  V  (d  r  •  v)dr  =  V  (d  r  •  v)  —  v. 

Substituting  : 

v=4v0  +  ^V(rfr.v)  +  !(Vxv)xdr.         (50) 

This  gives  another  form  of  (49)  which  is  sometimes  more 
convenient     It  is  also  slightly  more  symmetrical. 


164  VECTOR  ANALYSIS 

*  76.]  Consider  a  moving  fluid.  Let  v  (#,  yt  z,  f)  be  the 
velocity  of  the  fluid  at  the  point  (#,  y,  z)  at  the  time  t.  Sur- 
round a  point  (z0,  y^  z0)  with  a  small  sphere. 

d  r  •  d  r  =  c2. 

At  each  point  of  this  sphere  the  velocity  is 
v  =  v0  +  d  r  •  V  v. 

In  the  increment  of  time  8  1  the  points  of  this  sphere  will  have 

moved  the  distance 

(v0  +  d  r  •  V  v)  8  1. 

The  point  at  the  center  will  have  moved  the  distance 

v08t. 

The  distance  between  the  center  and  the  points  that  were 
upon  the  sphere  of  radius  d  r  at  the  commencement  of  the 
interval  8  t  has  become  at  the  end  of  that  interval  8  1 


To  find  the  locus  of  the  extremity  of  d  r'  it  is  necessary  to 
eliminate  d  r  from  the  equations 

dr'  =  dr  +  dr»  Vv  St, 
c2  =  d  i  •  d  r. 

The  first  equation  may  be  solved  for  d  r  by  the  method  of 
Art.  47,  page  90,  and  the  solution  substituted  into  the  second. 
The  result  will  show  that  the  infinitesimal  sphere 


has  been  transformed  into  an  ellipsoid  by  the  motion  of  the 
fluid  during  the  time  8  1. 

A  more  definite  account  of  the  change  that  has  taken  place 
may  be  obtained  by  making  use  of  equation  (50) 


THE  DIFFERENTIAL   CALCULUS   OF  VECTORS      165 


or  of  the  equation  (49) 

v  =  v0  +  V(dr.v)dr+(Vx  v)x  dr, 
v  =  v0  +  [V(dr.v)dr+|(Vxv)x  dr]+|(Vx  v)xdr. 

The  first  term  v0  in  these  equations  expresses  the  fact  that 
the  infinitesimal  sphere  is  moving  as  a  whole  with  an  instan- 
taneous velocity  equal  to  v0.  This  is  the  translational  element 
of  the  motion.  The  last  term 

|(Vxv)xdr  =  ^  curl  v  x  d  r 

shows  that  the  sphere  is  undergoing  a  rotation  about  an 
instantaneous  axis  in  the  direction  of  curl  v  and  with  an  angu- 
lar velocity  equal  in  magnitude  to  one  half  the  magnitude  of 
curl  v.  The  middle  term 


or  V(dr.v)rfr-(Vx  v)  x  dr 

expresses  the  fact  that  the  sphere  is  undergoing  a  defor- 
mation known  as  homogeneous  strain  by  virtue  of  which  it 
becomes  ellipsoidal.  For  this  term  is  equal  to 

d  x  V  v1  +  d  y  V  v2  -f  d  z  V  03, 

if  vv  vv  vs  be  respectively  the  components  of  v  in  the  direc- 
tions i,  j,  k.  It  is  fairly  obvious  that  at  any  given  point 
(#<»  y»i  zo)  a  set  of  three  mutually  perpendicular  axes  i,  j,  k 
may  be  chosen  such  that  at  that  point  Vvj,  Vv2,  Vv3  are  re- 


166  VECTOR  ANALYSIS 

spectively  parallel  to  them.      Then  the   expression   above 
becomes  simply 


The  point  whose  coordinates  referred  to  the  center  of  the 
infinitesimal  sphere  are 

d  x,    dy,    dz 

is  therefore  endowed  with  this,  velocity.     In  the  time  8  t  it 
will  have  moved  to  a  new  position 


The  totality  of  the  points  upon  the  sphere 


— -2 


goes  over  into  the  totality  of  points  upon  the  ellipsoid  of 
which  the  equation  is 

~2  «/2  «2 


The  statements  made  before  (Art.  72)  concerning  the  three 
types  of  motion  which  an  infinitesimal  sphere  of  fluid  may 
possess  have  therefore  now  been  demonstrated. 

77.]  The  symbolic  operator  V  may  be  applied  several  times 
in  succession.  This  will  correspond  in  a  general  way  to 
forming  derivatives  of  an  order  higher  than  the  first.  The 
expressions  found  by  thus  repeating  V  will  all  be  independ- 
ent of  the  axes  because  V  itself  is.  There  are  six  of  these 
dels  of  the  second  order. 

Let  F  (x,  y,  z)  be  a  scalar  function  of  position  in  space. 
The  derivative  V  V  is  a  vector  function  and  hence  has  a  curl 
and  a  divergence.  Therefore 

V.VF,  VxVF 


THE  DIFFERENTIAL   CALCULUS  OF   VECTORS      167 

are  the  two  derivatives  of  the  second  order  which  may  be 

obtained  from  V. 

V.VF=divVF  (51) 

VxVF=curlVF.  (52) 

The  second  expression  V  x  V  V vanishes  identically.  That  is, 
the  derivative  of  any  scalar  function  V  possesses  no  curl.  This 
may  be  seen  by  expanding  V  x  V  V  in  terms  of  i,  j,  k.  All 
the  terms  cancel  out.  Later  (Art.  83)  it  will  be  shown  con- 
versely that  if  a  vector  function  W  possesses  ho  curl,  i.  e.  if 

V  x  W=  curl  W  =  0,  then  W  =  VF, 

W  is  the  derivative  of  some  scalar  function  F. 

The  first  expression  V  •  V  F  when  expanded  in  terms  of 
i,  j,  k  becomes 


C/  C/  Cf 

Symbolically,          V  •  V  =  — ~  ^ »  H * 

The  operator  V  •  V  is  therefore  the  well-known  operator  of 
Laplace.     Laplace's  Equation 


= 0  (53) 

c/  ic~       ci  y~       &  z- 

becomes  in  the  notation  here  employed 

V.VF=0.  (53)' 

When  applied  to  a  scalar  function  F  the  operator  V  •  V  yields 
a  scalar  function  which  is,  moreover,  the  divergence  of  the 
derivative. 

Let  T  be  the  temperature  in  a  body.  Let  c  be  the  con- 
ductivity, p  the  density,  and  k  the  specific  heat.  The 
flow  f  is 


168  VECTOR  ANALYSIS 

The  rate  at  which  heat  is  leaving  a  point  per  unit  volume  per 
unit  time  is  V  •  f.     The  increment  of  temperature  is 


=-. 

p  K 

—  =-,v-vr. 

dt      pk 

This  is  Fourier's  equation  for  the  rate  of  change  of  tempera- 
ture. 

Let  V  be  a  vector  function,  and  Tv  Vv  Vz  its  three  com- 
ponents.   The  operator  V  •  V  of  Laplace  may  be  applied  to  V. 


(54) 

If  a  vector  function  V  satisfies  Laplace's  Equation,  each  of 
its  three  scalar  components  does.  Other  dels  of  the  second 
order  may  be  obtained  by  considering  the  divergence  and  curl 
of  V.  The  divergence  V  •  V  has  a  derivative 

VV.V  =  VdivV.  (55) 

The  curl  V  X  V  has  in  turn  a  divergence  and  a  curl, 

and  V-VxV,    VxVxV. 

V  •  V  x  V  =  div  curl  V  (56) 

and  V  x  V  x  V  =  curl  curl  V.  (57) 

Of  these  expressions  V  •  V  x  V  vanishes  identically.  That  is, 
the  divergence  of  the  curl  of  any  vector  is  zero.  This  may  be 
seen  by  expanding  V  •  V  x  V  in  terms  of  i,  j,  k.  Later  (Art. 
83)  it  will  be  shown  conversely  that  if  the  divergence  of  a 
vector  function  W  vanishes  identically,  i.  e.  if 

V  •  W  =  div  W  =  0,  then  .W  =  V  x  V  =  curl  V, 
W  is  the  curl  of  some  vector  function  V. 


THE  DIFFERENTIAL   CALCULUS  OF  VECTORS     169 

If  the  expression  V  x  (V  x  V)  were  expanded  formally 
according  to  the  law  of  the  triple  vector  product, 

Vx(VxV)=V«VV-V.VV. 

The  terms  V  •  VV  is  meaningless  until  V  be  transposed  to 
the  beginning  so  that  it  operates  upon  V. 

VxVxV  =  VV.V-V«VV,  (58) 

or  curl  curl  V  =  VdivV  —  V  •  V  V.  (58)' 

This  formula  is  very  important.  It  expresses  the  curl  of  the 
curl  of  a  vector  in  terms  of  the  derivative  of  the  divergence 
and  the  operator  of  Laplace.  Should  the  vector  function  V 
satisfy  Laplace's  Equation, 

V«VV  =  Oand 
curl  curl  V  =  V  div  V. 

Should  the  divergence  of  V  be  zero, 

curl  curl  V  =  —  V •  VV. 
Should  the  curl  of  the  curl  of  V  vanish, 
V  div  V  =  V  •  V  V. 
To  sum  up.     There  are  six  of  the  dels  of  the  second  order. 

V-VF,    VxVF, 

V-VV,    VV-V,     V-VxV,    VxVxV. 
Of  these,  two  vanish  identically. 

%  V  x  V  V  =  0,    V  •  V  x  V  =  0. 

A  third  may  be  expressed  in  terms  of  two  others. 

VxVxV  =  VV.V-V.VV.  (58) 

The  operator  V  •  V  is  equivalent  to  the  operator  of  Laplace. 


170  VECTOR  ANALYSIS 

*  78.]  The  geometric  interpretation  of  V«  Vw  is  interesting. 
It  depends  upon  a  geometric  interpretation  of  the  second 
derivative  of  a  scalar  function  u  of  the  one  scalar  variable  x. 
Let  ui  be  the  value  of  u  at  the  point  xt.  Let  it  be  required 
to  find  the  second  derivative  of  u  with  respect  to  x  at  the 
point  x0.  Let  xl  and  x2  be  two  points  equidistant  from  x0. 

That  is,  let 

x   —  x   =  x   —  x   =  a. 


Then  —  - 

is  the  ratio  of  the  difference  between  the  average  of  u  at  the 
points  x1  and  x2  and  the  value  of  u  at  x0  to  the  square  of  the 
distance  of  the  points  xv  x2  from  x0.  That 


ja 

d  u  2 


LiM         _ 
a==0          a* 

is  easily  proved  by  Taylor's  theorem. 

Let  u  be  a  scalar  function  of  position  in  space.  Choose 
three  mutually  orthogonal  lines  i,  j,  k  and  evaluate  the 
expressions 


Let  o!2  and  x1  be  two  points  on  the  line  i  at  a  distance  a  from 
x0  ;  a;4  and  xs,  two  points  on  j  at  the  same  distance  a  from 
x0  ;  xt  and  #6,  two  points  on  k  at  the  same  distance  a  from  x0. 


%    '-«, 

.  Lai     _J 


2  5»5 


_  LlM 

a=0 


THE  DIFFERENTIAL   CALCULUS  OF   VECTORS     171 


Add: 

2  2  2_ 


-  LDI  r 

a=OL 


As  V  and  V«  are  independent  of  the  particular  axes  chosen, 
this  expression  may  be  evaluated  for  a  different  set  of  axes, 
then  for  still  a  different  one,  etc.  By  adding  together  all 
these  results 

ii>\  +  w2  +  •  •  •  6  n  terms 


-w=  _  — 

6  X  a  =  0  2 


Let  n  become  infinite  and  at  the  same  time  let  the  different 
sets  of  axes  point  in  every  direction  issuing  from  x0.  The 

fraction 

u1  +  u2  +  ••  '  Qn  terms 
6n 

then  approaches  the  average  value  of  u  upon  the  surface  of  a 
sphere  of  radius  a  surrounding  the  point  x0.  Denote  this 
by  ««• 


V  •  V  u  is  equal  to  six  times  the  limit  approached  by  the  ratio 
of  the  excess  of  u  on  the  surface  of  a  sphere  above  the  value 
at  the  center  to  the  square  of  the  radius  of  the  sphere.  The 
same  reasoning  held  in  case  u  is  a  vector  function. 

If  u  be  the  temperature  of  a  body  V»Vw  (except  for  a 
constant  factor  which  depends   upon  the  material  of  the 


172  VECTOR  ANALYSIS 

body)  is  equal  to  the  rate  of  increase  of  temperature  (Art. 
77).  If  V»Vwis  positive  the  average  temperature  upon  a 
small  sphere  is  greater  than  the  temperature  at  the  center. 
The  center  of  the  sphere  is  growing  warmer.  In  the  case 
of  a  steady  flow  the  temperature  at  the  center  must  remain 
constant.  Evidently  therefore  the  condition  for  a  steady 

flow  is 

V  •  V  u  =  0. 

That  is,  the  temperature  is  a  solution  of  Laplace's  Equation. 

Maxwell  gave  the  name  concentration  to  —  V  •  V  u  whether 
u  be  a  scalar  or  vector  function.  Consequently  V  •  V  u  may 
be  called  the  dispersion  of  the  function  u  whether  it  be  scalar 
or  vector.  The  dispersion  is  proportional  to  the  excess  of 
the  average  value  of  the  function  on  an  infinitesimal  surface 
above  the  value  at  the  center.  In  case  u  is  a  vector  function 
the  average  is  a  vector  average.  The  additions  in  it  are 
vector  additions. 

SUMMARY  OF  CHAPTER  III 

If  a  vector  r  is  a  function  of  a  scalar  t  the  derivative  of 
r  with  respect  to  t  is  a  vector  quantity  whose  direction  is 
that  of  the  tangent  to  the  curve  described  by  the  terminus 
of  r  and  whose  magnitude  is  equal  to  the  rate  of  advance  of 
that  terminus  along  the  curve  per  unit  change  of  t.  The 
derivatives  of  the  components  of  a  vector  are  the  components 
of  the  derivatives. 

d^r_dnrl.      dn r2  .       dn rz  f 

TT»~~dl**  ~TrJ  +  Trk* 

A  combination  of  vectors  or  of  vectors  and  scalars  may  be 
differentiated  just  as  in  ordinary  scalar  analysis  except  that 
the  differentiations  must  be  performed  in  situ. 


THE  DIFFERENTIAL   CALCULUS  OF  VECTORS      173 

d  d  a  d  b 

_(..„),_.„  +  .._,  (S) 

d  d  a  d  b 


or  d  (a  •  b)  =  d  a  •  b  +  a  •  d  b,  (3)' 

d(axb)  =  daxb  +  axdb,  (4)' 

and  so  forth.  The  differential  of  a  unit  vector  is  perpendicu- 
lar to  that  vector. 

The  derivative  of  a  vector  r  with  respect  to  the  arc  s  of 
the  curve  which  the  terminus  of  the  vector  describes  is 
the  unit  tangent  to  the  curves  directed  toward  that  part  of  the 
curve  along  which  s  is  supposed  to  increase. 

£-*•  <•> 

The  derivative  of  t  with  respect  to  the  arc  a  is  a  vector  whose 
direction  is  normal  to  the  curve  on  the  concave  side  and 
whose  magnitude  is  equal  to  the  curvature  of  the  curve. 

dt      d*r 
C-d~s~d^' 

The  tortuosity  of  a  curve  in  space  is  the  derivative  of  the 
unit  normal  n  to  the  osculating  plane  with  respect  to  the 
arc  s. 


T  =  ^r=-^x^_i.  — =  ).        (ii) 

as       ds\ds      d  s*     v  C  •  C/ 
The  magnitude  of  the  tortuosity  is 

,  -  j-4  341  (13) 

cL  s   cL  s      d  s     I 

70  7         O 

/T     ff  ™  /"/     C  « 

(.to          (*  o 


174  VECTOR  ANALYSIS 

If  r  denote  the  position  of  a  moving  particle,  t  the  time, 
v  the  velocity,  A  the  acceleration, 

'-£-*  <15> 

•=£=*  <16> 

d  v      d2  r 


The  acceleration  may  be  broken  up  into  two  components  of 
which  one  is  parallel  to  the  tangent  and  depends  upon  the 
rate  of  change  of  the  scalar  velocity  v  of  the  particle  in  its 
path,  and  of  which  the  other  is  perpendicular  to  the  tangent 
and  depends  upon  the  velocity  of  the  particle  and  the  curva- 

ture of  the  path. 

A  =  s  t  +  vz  C.  (19) 

Applications  to  the  hodograph,  in  particular  motion  in  a 
circle,  parabola,  or  under  a  central  acceleration.  Application 
to  the  proof  of  the  theorem  that  the  motion  of  a  rigid  body 
one  point  of  which  is  fixed  is  an  instantaneous  rotation  about 
an  axis  through  the  fixed  point. 

Integration  with  respect  to  a  scalar  is  merely  the  inverse 
of  differentiation.  Application  to  finding  the  paths  due  to 
given  accelerations. 

The  operator  V  applied  to  a  scalar  function  of  position  in 
space  gives  a  vector  whose  direction  is  that  of  most  rapid 
increase  of  that  function  and  whose  magnitude  is  equal  to 
the  rate  of  that  increase  per  unit  change  of  position  in  that 
direction 


<22> 


THE  DIFFERENTIAL    CALCULUS   OF   VECTORS     175 

The  operator  V  is  invariant  of  the  axes  i,  j,  k.    It  may  be 
defined  by  the  equation 

«,  (24) 


or  VV.dr  =  dV.  (25)' 

Computation  of  the  derivative  V  V  by  two  methods  depend- 
ing upon  equations  (21)  and  (25)'.  Illustration  of  the  oc- 
currence of  V  in  mathematical  physics. 

V  may  be  looked  upon  as  a  fictitious  vector,  a  vector 
differentiator.  It  obeys  the  formal  laws  of  vectors  just  in 
so  far  as  the  scalar  differentiators  of  9/  9  #,  9  /  5  y,  9[  dz  obey 
the  formal  laws  of  scalar  quantities 


If  a  be  a  unit  vector  a  •  V  V  is  the  directional  derivative  of  V 
in  the  direction  a. 

a.VF=(a«V)  F=  a  •  (VF).  (30) 

If  V  is  a  vector  function  a  •  V  V  is  the  directional  derivative 
of  that  vector  function  in  the  direction  a. 


+  J.        +  *.  (32)' 

d  x          9  y  9  z 


=ix        +  jx        +  kx,        (33)' 

d  X  3  d  Z 


176  VECTOR  ANALYSIS 

Proof  that  V  •  V  is  the  divergence  of  V  and  V  x  V,  the  curl 

of  V. 

V  •  V  =  div  V, 

V  x  V  =  curl  V. 

V  (w  4-  v)  =  V  u  +  V0,  (35) 

V  •  (u  +  v)  =  V.  u  +  V-  v,  (36) 

V  x  (u  +  v)  =  V  x  u  +  V  x  v,  (37) 

V  (u  v~)  —  v  V  u  +  w  V  v,  (38) 

V  •  (w  v)  =  V  w  •  v  +  u  V  •  v,  (39) 

Vx(wv)  =  Vwxv  +  wVxv,  (40) 

V(u.v)=vVu  +  ii»Vv  +  vx  (Vxu) 

+  ux  (Vxv),  (41) 

V»(uxv)=vVxn  -  u.Vxv,        (42) 
Vx  (u  x  v)=v«Vu  —  v  V-  u-n  -Vv  +  uV»  v.  (43) 

Introduction  of  the  partial  del,  V  (u  •  v)n,  in  which  the  dif- 
ferentiations are  performed  upon  the  hypothesis  that  u  is 

constant. 

nx  (Vxv)=V(u»  v)u-n«Vv.  (46) 


If  a  be  a  unit  vector  the  directional  derivative 

a  •  V  v  =  V  (a  •  v)a  +  (V  x  v)  x  a.          (47) 

The  expansion  of  any  vector  function  v  in  the  neighborhood 
of  a  point  (a;,,,  y#  «0)  at  which  it  takes  on  the  value  of  v0  is 

v  =  v0  +  V  (d  r  •  v)dr  +  (V  x  v)  x  rfr,       (49) 
or  v  =  |v0  +  V(^r.v)+^(Vxv)  X  dt.       (50) 

Application  to  hydrodynamics. 

The  dels  of  the  second  order  are  six  in  number. 


THE  DIFFERENTIAL   CALCULUS   OF   VECTORS      177 
VxVF=curlVF=0,  (52) 

T2F"      £)2V       32F" 

V.vrdivVF-f^+f^  +  l^      (51) 

V  •  V  is  Laplace's  operator.     If  V^VF=0,  V  satisfies  La- 
place's Equation.     The  operator  may  be  applied  to  a  vector. 


VV  •  V  =  V  div  V,  (55) 

V  •  V  x  V  =  div  curl  V  =  (7,  (56) 

V  x  V  x  V  =  curl  curl  V  =  VV.V-V.VV.   (58) 

The  geometric  interpretation  of  V  •  V  as  giving  the  disper- 
sion of  a  function. 

EXERCISES  ON  CHAPTER  III 

1.  Given  a  particle  moving  in  a  plane  curve.    Let  the 
plane  be  the  ij-plane.     Obtain  the  formulae  for  the  compo- 
nents of  the  velocity  parallel  and  perpendicular  to  the  radius 
vector  r.     These  are 

r-,    6  k  X  r, 

where  6  is  the  angle  the  radius  vector  r  makes  with  i,  and  k 
is  the  normal  to  the  plane. 

2.  Obtain  the  accelerations  of  the  particle  parallel  and 
perpendicular  to  the  radius  vector.     These  are 

(r_r02)-,     (r  0+  2r  0)  k  X-- 
r  r 

Express  these  formulae  in  the  usual  manner  in  terms  of  x 
and  y. 

12 


178  VECTOR  ANALYSIS 

3.  Obtain  the  accelerations  of  a  moving  particle  parallel 
and  perpendicular  to  the  tangent  to  the  path  and  reduce  the 
results  to  the  usual  form. 

4.  If  r,  <£,  6  be  a  system  of  polar  coordinates  in  space, 
where  r  is  the  distance  of  a  point  from  the  origin,  <f>  the 
meridianal  angle,  and  6  the  polar  angle ;  obtain  the  expressions 
for  the  components  of  the  velocity  and  acceleration  along  the 
radius  vector,  a  meridian,  and  a  parallel  of  latitude.     Reduce 
these  expressions  to  the  ordinary  form  in  terms  of  #,  y,  z. 

5.  Show  the  direct  method  suggested  in  Art.  63  that  the 
operator  V  is  independent  of  the  axes. 

6.  By  the  second  method  given   for   computing   V  find 
the  derivative  V  of  a  triple  product  [a  b  c]  each  term  of  which 
is  a  function  of  a;,  #,  z  in  case 

a  =  (r  •  r)  r,      b  =  (r  •  a)  e,      c  =  r  X  t, 

where  d,  *e,  f  are  constant  vectors. 

7.  Compute  V«  V  Fwhen  Fis  r2,      r,       -,      or  —  • 

r  r2 

8.  Compute  V  •  VV,  VV  .  V,  and  V  x  V  x  V  when  V  is 

equal  to  r  and  when  V  is  equal  to  -?  and  show  that  in  these 

r3 
cases  the  formula  (58)  holds. 

9.  Expand  V  x  VF  and  V  •  V  x  V  in  terms  of  i,  j,  k  and 
show  that  they  vanish  (Art.  77). 

10.  Show  by  expanding  in  terms  of  i,  j,  k  that 

Vx  VxV=VV.V-V.  VV. 

11.  Prove    A.V(V«W)  =  VA.  VW+WA- VV, 
and 

(VxV)  x  W  =  Vx  (Vx  W)w 


CHAPTER  IV 

THE  INTEGRAL  CALCULUS   OF  VECTORS 

79.]  Let  W  (z,  y,  z)  be  a  vector  function  of  position  in 
space.  Let  C  be  any  curve  in  space,  and  r  the  radius  vector 
drawn  from  some  fixed  origin  to  the  points  of  the  curve. 
Divide  the  curve  into  infinitesimal  elements  dr.  From  the 
sum  of  the  scalar  product  of  these  elements  d  r  and  the  value 
of  the  function  W  at  some  point  of  the  element  — 


thus 


2  W  •  d  r. 


The  limit  of  this  sum  when  the  elements  dr  become  infinite 
in  number,  each  approaching  zero,  is  called  the  line  integral  of 
W  along  the  curve  0  and  is  written 


-  dr. 


If 

and 


fw«  dr=  C 
J  c  Jo 


(1) 


The  definition  of  the  line  integral  therefore  coincides  with 
the  definition  usually  given.  It  is  however  necessary  to 
specify  in  which  direction  the  radius  vector  r  is  supposed  to 
describe  the  curve  during  the  integration.  For  the  elements 
d  r  have  opposite  signs  when  the  curve  is  described  in  oppo- 


180  VECTOR  ANALYSIS 

site  directions.     If  one  method  of  description  be  denoted  by 
C  and  the  other  by  —  (7, 


/W  'dr  =  —  I 
-o  J 


W*  dr. 
o 


In  case  the  curve  C  is  a  closed  curve  bounding  a  portion  of 
surface  the  curve  will  always  be  regarded  as  described  in 
such  a  direction  that  the  enclosed  area  appears  positive 
(Art.  25). 

If  f  denote  the  force  which  may  be  supposed  to  vary  from 
point  to  point  along  the  curve  (7,  the  work  done  by  the  force 
when  its  point  of  application  is  moved  from  the  initial  point 
r0  of  the  curve  C  to  its  final  point  r  is  the  line  integral 


Cf.dr  =   Ff.  dr. 

JO  J  r0 


Theorem  :  The  line  integral  of  the  derivative  V  V  of  a 
scalar  function  F(z,  y,  z)  along  any  curve  from  the  point 
r0  to  the  point  r  is  equal  to  the  difference  between  the  values 
of  the  function  V  "(#,  y,  z)  at  the  point  r  and  at  the  point  r0. 
That  is, 


=  F(r)  -  F(r0)  =  F(*f  y,  «)  -  V  (»„  yw  O- 
By  definition  d  r  .  V  F=  d  V 

r* 

/  dV=  F(r)  -  F(r.)  =  F(a,y,«)  -  F(*0,y0,<).    (2) 

v  r 

o 

Theorem,  :    The  line  integral   of  the   derivative  V  F  of  a 
ifi^     single  valued  scalar  function  of  position   V  taken  around  a 
closed  curve  vanishes. 

The  fact  that  the  integral  is  taken  around  a  closed  curve 
is  denoted  by  writing  a  circle  at  the  foot  of  the  integral  sign. 
To  show  « 

r  =  0.  (3) 


THE  INTEGRAL   CALCULUS  OF  VECTORS          181 
The  initial  point  r0  and  the  final  point  r  coincide.     Hence 

F(r)  =  F(r0). 


Hence  by  (2)  f 

Jo 


Theorem :  Conversely  if  the  line  integral  of  W  about  every 
closed  curve  vanishes,  W  is  the  derivative  of  some  scalar 
function  F  (x,  y,  z)  of  position  in  space. 

Given  I  W  •  d  r  =  0. 

*/  O 

To  show  W  =  V  F. 

Let  r0  be  any  fixed  point  in  space  and  r  a  variable  point. 
The  line  integral 

f  W-rfr 

o 

is  independent  of  the  path  of  integration  G.  For  let  any  two 
paths  G  and  G'  be  drawn  between  r0  and  r.  The  curve  which 
consists  of  the  path  G  from  r0  to  r  and  the  path  —  G'  from  r 
to  r0  is  a  closed  curve.  Hence  by  hypothesis 

;  r  r 

v  c  v  —G' 

/F 
-G'  *J  c' 

Hence  iw*dr  =    fw»dr. 

J  c  J  c' 

Hence  the  value  of  the  integral  is  independent  of  the  path 
of  integration  and  depends  only  upon  the  final  point  r. 


182  VECTOR  ANALYSIS 

The  value  of  the  integral  is  therefore  a  scalar  function  of 
the  position  of  the  point  r  whose  coordinates  are  a:,  y,  z. 


f. 


W  •  d  r  =  V  O,  y,  2). 


Let  the  integral  be  taken  between  two  points  infinitely  near 
together. 


But  by  definition  V  V*  d  r  =  d  V. 

Hence  W 


The  theorem  is  therefore  demonstrated. 

80.]  Let  f  be  the  force  which  acts  upon  a  unit  mass  near 
the  surface  of  the  earth  under  the  influence  of  gravity.  Let 
a  system  of  axes  i,  j,  k  be  chosen  so  that  k  is  vertical.  Then 


The  work  done  by  the  force  when  its  point  of  application 
moves  from  the  position  r0  to  the  position  r  is 


w 


—    I  f»dr  =    f  —  gk»dr  =  —  f  g  dz. 

«/  r  J  r  «/ r 


Hence  w  =  -  g  (z  —  z0)  =  g  (ZQ  —  z). 

The  force  f  is  said  to  be  derivable  from  a  force-function  V 
when  there  exists  a  scalar  function  of  position  V  such  that 
the  force  is  equal  at  each  point  of  the  derivative  VF". 
Evidently  if  F  is  one  force-function,  another  may  be  obtained 
by  adding  to  V  any  arbitrary  constant.  In  the  above  ex- 
ample the  force-function  is 

V=w=g(z0-e). 
Or  more  simply  V  —  —  g  z. 

The  force  is  f  =  VF=  -    k. 


THE  INTEGRAL   CALCULUS  OF  VECTORS         183 

The  necessary  and  sufficient  condition  that  a  force-function 
V  (x,  y,  z)  exist,  is  that  the  work  done  by  the  force  when  its 
point  of  application  moves  around  a  closed  circuit  be  zero. 

The  work  done  by  the  force  is 

w  =  I  f'di. 


If  this  integral  vanishes  when  taken  around  every  closed 
contour  f  =  VF==VM; 

And  conversely  if  f  =  V  V 

the  integral  vanishes.     The  force-function  and  the  work  done 

differ  only  by  a  constant. 

V  =  w  +  const. 

In  case  there  is  friction  no  force-function  can  exist.  For  the 
work  done  by  friction  when  a  particle  is  moved  around  in  a 
closed  circuit  is  never  zero. 

The  force  of  attraction  exerted  by  a  fixed  mass  M  upon 
a  unit  mass  is  directed  toward  the  fixed  mass  and  is  propor- 
tional to  the  inverse  square  of  the  distance  between  the 

masses. 

M 

f  =  -  c  —  r. 
rs 

This  is  the  law  of  universal  gravitation  as  stated  by  Newton. 
It  is  easy  to  see  that  this  force  is  derivable  from  a  force- 
function  V.  Choose  the  origin  of  coordinates  at  the  center 
of  the  attracting  mass  M.  Then  the  work  done  is 

rr    M 
w  =  —  I     c  -r-  T»dr. 

Jr       rs 

o 

But  r  •  d  r  =  r  d  r, 


„    C 
=  —  cM 

J  r 


—-——cM 
2 


184  VECTOR  ANALYSIS 

By  a  proper  choice  of  units  the  constant  c  may  be  made 
equal  to  unity.  The  force-function  V  may  therefore  be 
chosen  as 


If  there  had  been  several  attracting  bodies  Mv  Mv 
the  force-function  would  have  been 


y  __ 


where  rv  r2,  r3,  •  •  •  are  the  distances  of  the  attracted  unit 
mass  from  the  attracting  masses  Mv  Mz,  Ms  -  •  • 

The  law  of  the  conservation  of  mechanical  energy  requires 
that  the  work  done  by  the  forces  when  a  point  is  moved 
around  a  closed  curve  shall  be  zero.  This  is  on  the  assump- 
tion that  none  of  the  mechanical  energy  has  been  converted 
into  other  forms  of  energy  during  the  motion.  The  law  of 
conservation  of  energy  therefore  requires  the  forces  to  be 
derivable  from  a  force-function.  Conversely  if  a  force- 
function  exists  the  work  done  by  the  forces  when  a  point  is 
carried  around  a  closed  curve  is  zero  and  consequently  there 
is  no  loss  of  energy.  A  mechanical  system  for  which  a  force- 
function  exists  is  called  a  conservative  system.  From  the 
example  just  cited  above  it  is  clear  that  bodies  moving  under 
the  law  of  universal  gravitation  form  a  conservative  system  — 
at  least  so  long  as  they  do  not  collide. 

81.]  Let  W  (x,  y,  2)  be  any  vector  function  of  position  in 
space.  Let  S  be  any  surface.  Divide  this  surface  into  in- 
finitesimal elements.  These  elements  may  be  regarded  as 
plane  and  may  be  represented  by  infinitesimal  vectors  of 
which  the  direction  is  at  each  point  the  direction  of  the 
normal  to  the  surface  at  that  point  and  of  which  the  magni- 
tude is  equal  to  the  magnitude  of  the  area  of  the  infinitesimal 


THE  INTEGRAL   CALCULUS  OF  VECTORS          185 

element.  Let  this  infinitesimal  vector  which  represents  the 
element  of  surface  in  magnitude  and  direction  be  denoted  by 
d  a.  Form  the  sum 


which  is  the  sum  of  the  scalar  products  of  the  value  of  W 
at  each  element  of  surface  and  the  (vector)  element  of 
surface.  The  limit  of  this  sum  when  the  elements  of  sur- 
face approach  zero  is  called  the  surface  integral  of  W  over 
the  surface  S,  and  is  written 


// 


""• 


The  value  of  the  integral  is  scalar.     If  W  and  d&  be  ex- 
pressed in  terms  of  their  three  components  parallel  to  i,  j,  k 

W  =  Wl  i  +  Wz  j  +  Wz  k, 
d  a  =  (d  a  •  i)  i  +  (d  a  •  j)  j  +  (d  a  •  k)  k, 
or  da,  =  dy  dz  i  +  dzdx  j  +  dxdy  k, 

(5) 


The  surface  integral  therefore  has  been  defined  as  is  cus- 
tomary in  ordinary  analysis.  It  is  however  necessary  to 
determine  with  the  greatest  care  which  normal  to  the  surface 
d  a  is.  That  is,  which  side  of  the  surface  (so  to  speak)  the 
integral  is  taken  over.  For  the  normals  upon  the  two  sides 
are  the  negatives  of  each  other.  Hence  the  surface  integrals 
taken  over  the  two  sides  will  differ  in  sign.  In  case  the 
surface  be  looked  upon  as  bounding  a  portion  of  space  d  a 
is  always  considered  to  be  the  exterior  normal. 

If  f  denote  the  flux  of  any  substance  the  surface  integral 

f-i» 

8 


186  VECTOR  ANALYSIS 

gives  the  amount  of  that  substance  which  is  passing  through 
the  surface  per  unit  time.  It  was  seen  before  (Art.  71)  that 
the  rate  at  which  matter  was  leaving  a  point  per  unit 
volume  per  unit  time  was  V  •  f.  The  total  amount  of  mat- 
ter which  leaves  a  closed  space  bounded  by  a  surface  £  per 
unit  time  is  the  ordinary  triple  integral 

V*tdv.  (6) 

Hence  the  very  important  relation  connecting  a  surface  in- 
tegral of  a  flux  taken  over  a  closed  surface  and  the  volume 
integral  of  the  divergence  of  the  flux  taken  over  the  space 
enclosed  by  the  surface  — 


=   fff 


(7) 


Written  out  in  the  notation  of  the  ordinary  calculus  this 
becomes 

\f[Xdydz  +  Tdzdx  +  Zdxdy] 


where  X,  F,  Z  are  the  three  components  of  the  flux  f  .  The 
theorem  is  perhaps  still  more  familiar  when  each  of  the  three 
components  is  treated  separately. 


ff  X  dx  dy  =  fff  *^  dxdy  dz.  (8)' 

This  is  known  as  Gauss's  Theorem.  It  states  that  the  surface 
integral  (taken  over  a  closed  surface)  of  the  product  of  a 
function  X  and  the  cosine  of  the  angle  which  the  exterior 
normal  to  that  surface  makes  with  the  X-axis  is  equal  to 
the  volume  integral  of  the  partial  derivative  of  that  function 


THE  INTEGRAL   CALCULUS  OF  VECTORS          187 

with  respect  to  x  taken  throughout  the  volume  enclosed  by 
that  surface. 

If  the  surface  S  be  the  surface  bounding  an  infinitesimal 
sphere  or  cube 

CC  f.d&  =  V.f  dv 

where  d  v  is  the  volume  of  that  sphere  or  cube.    Hence 


.  (9) 

dv  J  J  a 

This  equation  may  be  taken  as  a  definition  of  the  divergence 
V  •  f.  The  divergence  of  a  vector  function  f  is  equal  to  the 
limit  approached  by  the  surface  integral  of  f  taken  over  a  sur- 
face bounding  an  infinitesimal  body  divided  by  that  volume 
when  the  volume  approaches  zero  as  its  limit.  That  is 


From  this  definition  which  is  evidently  independent  of  the 
axes  all  the  properties  of  the  divergence  may  be  deduced.  In 
order  to  make  use  of  this  definition  it  is  necessary  to  develop 
at  least  the  elements  of  the  integral  calculus  of  vectors  before 
the  differentiating  operators  can  be  treated.  This  definition 
of  V  •  f  consequently  is  interesting  more  from  a  theoretical 
than  from  a  practical  standpoint. 

82.]  Theorem  :  The  surface  integral  of  the  curl  of  a  vector 
function  is  equal  to  the  line  integral  of  that  vector  function 
taken  around  the  closed  curve  bounding  that  surface. 

f/Vx  W-rfa=  Cw-dr.  (11) 

«/  «/  g  *J  o 

This  is  the  celebrated  theorem  of  Stokes.  On  account  of  its 
great  importance  in  all  branches  of  mathematical  physics  a 
number  of  different  proofs  will  be  given. 


188 


VECTOR  ANALYSIS 


First  Proof :  Consider  a  small  triangle  1 23  upon  the  surface 
S  (Fig.  32).  Let  the  value  of  W  at  the  vertex  1  be  W0. 
Then  by  (50),  Chap.  III.,  the  value  at  any  neighboring  point  is 

W  =  \\  W0  +  V  (W  •  B  r)  +  (V  x  W)  x  8  r   j , 

where  the  symbol  S  r  has  been  introduced  for  the  sake  of  dis- 
tinguishing it  from  d  r  which  is  to  be  used  as  the  element  of 
integration.  The  integral  of  W  taken  around  the  triangle 


FIG.  32. 


Cw-di=l   fw 

t/A  «/  A 

+  ~  f  (V  x  W)  x  Sr-dr. 

*J  A 

Thefirstterm         \   Cwo.dr  =  ±W0.   f 

«/  A  t/ 


dr 


vanishes  because  the  integral  of  d  r  around  a  closed  figure,  in 
this  case  a  small  triangle,  is  zero.     The  second  term 


vanishes  by  virtue  of  (3)  page  180.    Hence 


THE  INTEGRAL   CALCULUS  OF   VECTORS         189 


=l-    f 

z  J  A 


A 

Interchange  the  dot  and  the  cross  in  this  triple  product 
fw«dr=4   /VxW'Srxdr. 

J  A  «/  A 

When  d  r  is  equal  to  the  side  1 2  of  the  triangle,  B  r  is  also 
equal  to  this  side.  Hence  the  product 

BT  x  di 

vanishes  because  8  r  and  d  r  are  collinear.  In  like  manner 
when  di  is  the  side  31,  Sr  is  the  same  side.7#,  but  taken 
in  the  opposite  direction.  Hence  the  vector  product  vanishes. 
When  d  r  is  the  side  23,  S  r  is  a  line  drawn  from  the  vertex 
1  at  which  W= W0  to  this  side  23.  Hence  the  product  8  r  x  d  r 
is  twice  the  area  of  the  triangle.  This  area,  moreover,  is  the 
positive  area  123.  Hence 

2$r  x  di  =  d a, 

where  d  a  denotes  the  positive  area  of  the  triangular  element 
of  surface.  For  the  infinitesimal  triangle  therefore  the 

relation  /» 

I  W*dr  =  V  x  W«  d* 

J  A 

holds. 

Let  the  surface  S  be  divided  into  elementary  triangles. 
For  convenience  let  the  curve  which  bounds  the  surface 
be  made  up  of  the  sides  of  these  triangles.  Perform  the 
integration 


A 

around  each  of  these  triangles  and  add  the  results  together. 


190  VECTOR  ANALYSIS 

The  second  member      2  ^  x  ^  * d  a 

-8 

is  the  surface  integral  of  the  curl  of  W. 


In  adding  together  the  line  integrals  which  occur  in  the  first 
member  it  is  necessary  to  notice  that  all  the  sides  of  the  ele- 
mentary triangles  except  those  which  lie  along  the  bounding 
curve  of  the  surface  are  traced  twice  in  opposite  directions. 
Hence  all  the  terms  in  the  sum 


which  arise  from  those  sides  of  the  triangles  lying  within  the 
surface  S  cancel  out,  leaving  in  the  sum  only  the  terms 
which  arise  from  those  sides  which  make  up  the  bounding 
curve  of  the  surface.  Hence  the  sum  reduces  to  the  line  in- 
tegral of  W  along  the  curve  which  bounds  the  surface  S. 


Hence 


V  x 


=/w 

t/o 


=  1  W.dT. 


FIG.  33. 


Second  Proof :  Let  C  be  any  closed 
contour  drawn  upon  the  surface  S 
(Fig.  33).  It  will  be  assumed  that  C 
is  continuous  and  does  not  cut  itself. 
Let  C'  be  another  such  contour  near 
to  C.  Consider  the  variation  S  which 
takes  place  in  the  line  integral  of  W 
in  passing  from  the  contour  C  to  the 
contour  C". 


THE  INTEGRAL   CALCULUS  OF  VECTORS         191 

r         c         c 

J  J  C'  Jc 

/r  r  c 

W  •  d  r  =  I  8  (W  •c£r)=  I  W  •  S  d  r  +  I  oW •  dr. 
J  J  J 

and  8  d  r  =  d  S  r. 

Hence   /  W  •  8  d  r  =  /W«dSr=/d(W»8r)  —  I  dW  *ST. 

The  expression  d  (W  •  S  r)  is  by  its  form  a  perfect  differential. 
The  value  of  the  integral  of  that  expression  will  therefore  be 
the  difference  between  the  values  of  W  •  d  r  at  the  end  and  at 
the  beginning  of  the  path  of  integration.  In  this  case  the 
integral  is  taken  around  the  closed  contour  C.  Hence 

f* 
Jc 

//» 
W  •  8  d  r  =  —  I  d  W  •  8  r, 

r  r  r 

and  S  I  W*di=  I  SW  •  dr  —  I^W»8r, 

8  iw »di=  C\  SW»dr  —  dW*Sil. 

9W  9W  9W 

But  d  W  =  -= —  d  x  +  -= —  d  y  +  -z —  d  2, 

d  x  ay  <y  z 


9W  9W  .  ^ 

or  d  W  =  -= —  i  •  d  r  +  -^ —  J  •  d  r  +  -^—  k  •  d  r, 

d  x  d  y  <y  z 

9  W^  9  W^  9     ^ 

and 


192  VECTOR  ANALYSIS 

Substituting  these  values 

/r  (  9  W  9  W 

W-dr=t  \-^--dr  i-Sr-- 
J   (  3  x  ax 

+  similar  terms  in  y  and  z.  | 
But  by  (25)  page  111 


3  W\  3  W  5  W 

r  x  d  r)  =  -=—  •  d  r  i  •  3  r  —  -^  —  •  5  r  i  •  d  r. 

3  x  3  x 


(3  W\ 
i  x  •=  —  j 
d  x  J 

/C  (        3  W 
W'di=J   jix-^—  •Srxdr 

+  similar  terms  in  y  and  2  |  . 
8  f 


or 


In  Fig.  33  it  will  be  seen  that  d  r  is  the  element  of  arc 
along  the  curve  0  and  S  r  is  the  distance  from  the  curve  C  to 
the  curve  C".  Hence  S  r  x  d  r  is  equal  to  the  area  of  an  ele- 
mentary parallelogram  included  between  C  and  C'  upon  the 
surface  S.  That  is 

Br  X  di  =  d&, 

S  Cw*dr=  TV  x  W«da. 

Let  the  curve  #  starting  at  a  point  0  in  $  expand  until  it 
coincides  with  the  contour  bounding  S.     The  line  integral 

fW'di 

will  vary  from  the  value  o  at  the  point  0  to  the  value 

IW-dr 

Jo 


THE  INTEGRAL   CALCULUS  OF  VECTORS         193 

taken  around  the  contour  which  bounds  the  surface  S.  This 
total  variation  of  the  integral  will  be  equal  to  the  sum  of  the 
variations  8 


if 


Or 

83.]  Stokes's  theorem  that  the  surface  integral  of  the  curl 
of  a  vector  function  is  equal  to  the  line  integral  of  the  func- 
tion taken  along  the  closed  curve  which  bounds  the  surface 
has  been  proved.  The  converse  is  also  true.  If  the  surface 
integral  of  a  vector  function  U  is  equal  to  the  line  integral  of  the 
function  W  taken  around  the  curve  bounding  the  surface  and  if 
this  relation  holds  for  all  surfaces  in  space,  then  U  is  the  curl  of 
W.  That  is 

f  fu.rfa  =   Cw'di,  thenTJ=Vx  W.       (12) 

Form  the  surface  integral  of  the  difference  between  TT  and 
V  x  W. 

f  C  (TJ-Vx  W)»da  =  f  W«dr  -    f  W«rfr  =  0, 

or  f  f  (U-Vx  W)-da  =  0. 

Let  the  surface  S  over  which  the  integration  is  performed  be 
infinitesimal.     The  integral  reduces  to  merely  a  single  term 

(U  -  V  x  W)  •  d  a  =  0. 

As  this  equation  holds  for  any  element  of  surface  d  a,  the 
first  factor  vanishes.     Hence 

U  -  V  x  W  =  0. 

Hence  U  =  V  x  W. 

The  converse  is  therefore  demonstrated. 

13 


194  VECTOR  ANALYSIS 

A  definition  of  V  x  W  which  is  independent  of  the  axes 
i,  j,  k  may  be  obtained  by  applying  Stokes's  theorem  to  an  in- 
finitesimal plane  area.  Consider  a  point  P.  Pass  a  plane 
through  P  and  draw  in  it,  concentric  with  P,  a  small  circle  of 
area  d  a. 

Vx  W-da  =  f  W»dr.  (13) 

Jo 

When  d  a  has  the  same  direction  as  V  x  W  the  value  of  the 
line  integral  will  be  a  maximum,  for  the  cosine  of  the  angle 
between  V  x  W  and  d  a  will  be  equal  to  unity.  For  this 
value  of  da, 


r 

Jo 


Hence  the  curl  V  x  W  of  a  vector  function  W  has  at  each 
point  of  space  the  direction  of  the  normal  to  that  plane  in 
which  the  line  integral  of  W  taken  about  a  small  circle  con- 
centric with  the  point  in  question  is  a  maximum.  The  mag- 
nitude of  the  curl  at  the  point  is  equal  to  the  magnitude  of 
that  line  integral  of  maximum  value  divided  by  the  area  of 
the  circle  about  which  it  is  taken.  This  definition  like  the 
one  given  in  Art.  81  for  the  divergence  is  interesting  more 
from  theoretical  than  from  practical  considerations. 

Stokes's  theorem  or  rather  its  converse  may  be  used  to  de- 
duce Maxwell's  equations  of  the  electro-magnetic  field  in  a 
simple  manner.  Let  E  be  the  electric  force,  B  the  magnetic 
induction,  H  the  magnetic  force,  and  C  the  flux  of  electricity 
per  unit  area  per  unit  time  (i.  e.  the  current  density). 

It  is  a  fact  learned  from  experiment  that  the  total  electro- 
motive force  around  a  closed  circuit  is  equal  to  the  negative 
of  the  rate  of  change  of  total  magnetic  induction  through 
the  circuit.  The  total  electromotive  force  is  the  line  integral 
of  the  electric  force  taken  around  the  circuit.  That  is 


THE  INTEGRAL   CALCULUS  OF  VECTORS         195 

The  total  magnetic  induction  through  the  circuit  is  the  sur- 
face integral  of  the  magnetic  induction  B  taken  over  a  surface 
bounded  by  the  circuit.  That  is 


3 

Experiment  therefore  shows  that 

/d      C  C  „ 
E.rfr  =  — —     I    i   B.rfa, 
j  a  t   J  J  a 

or  I  E*dr=   I  I   —  B  •  d  a. 

Jo  J  J  a 

Hence  by  the  converse  of  Stokes's  theorem 

V  x  E  =  —  B,     curl  E  =  —  B. 

It  is  also  a  fact  of  experiment  that  the  work  done  in  carry- 
ing a  unit  positive  magnetic  pole  around  a  closed  circuit  is 
equal  to  4?r  times  the  total  electric  flux  through  the  circuit. 
The  work  done  in  carrying  a  unit  pole  around  a  circuit  is 
the  line  integral  of  H  around  the  circuit.  That  is 


The  total  flux  of  electricity  through  the  circuit  is  the 
surface  integral  of  C  taken  over  a  surface  bounded  by  the 
circuit.  That  is 


Experiment  therefore  teaches  that 


rH.rfr  =  4?r    IJ  (NcZa. 


196  VECTOR  ANALYSIS 

By  the  converse  of  Stokes's  theorem 
V  x  H  =  4  TT  C. 

With  a  proper  interpretation  of  the  current  C,  as  the  dis- 
placement current  in  addition  to  the  conduction  current, 
an  interpretation  depending  upon  one  of  Maxwell's  primary 
hypotheses,  this  relation  and  the  preceding  one  are  the  funda- 
mental equations  of  Maxwell's  theory,  in  the  form  used  by 
Heaviside  and  Hertz. 

The  theorems  of  Stokes  and  Gauss  may  be  used  to  demon- 
strate the  identities. 

V .  V  x  W  =  0,      div  curl  W  =  0. 

VxVF=0,      curlVF=0. 
According  to  Gauss's  theorem 

fff  V.Vx  W<20  =    ffvxW.da. 
J  J  J  J  J  s 

According  to  Stokes's  theorem 

X  Wtd*=    f  W-dr. 
Hence  C  C  C  V .  V  x  W  d?  =    Cw*dr. 

Apply  this  to  an  infinitesimal  sphere.  The  surface  bounding 
the  sphere  is  closed.  Hence  its  bounding  curve  reduces  to  a 
point ;  and  the  integral  around  it,  to  zero. 

V.VxWdv=    fw.<fr  =  0, 
Jo 

V»V  x  W  =  0. 


THE  INTEGRAL  CALCULUS  OF  VECTORS          197 

Again  according  to  Stokes's  theorem 

ffvxVF.  da=    fvF-dr. 

J  J  3  J  0 

Apply  this  to  any  infinitesimal  portion  of  surface.  The  curve 
bounding  this  surface  is  closed.  Hence  the  line  integral  of 
the  derivative  VF"  vanishes. 


V  x 

As  this  equation  holds  for  any  d  a,  it  follows  that 

Vx  VF=0. 

In  a  similar  manner  the  converse  theorems  may  be 
demonstrated.  If  the  divergence  V  •  U  of  a  vector  function 
U  is  everywhere  zero,  then  U  is  the  curl  of  some  vector 

function  W. 

U  =  V  x  W. 

If  the  curl  V  x  U  of  a  vector  function  U  is  everywhere  zero, 
then  U  is  the  derivative  of  some  scalar  function  F", 


84.]  By  making  use  of  the  three  fundamental  relations 
between  the  line,  surface,  and  volume  integrals,  and  the 
dels,  viz.  : 


f  VF.rfr  =  F(r)-F(r0),  (2) 

•''o 

ffvxW»rfa  =    fw.rfr,  (11) 


it  is  possible  to  obtain  a  large  number  of  formulae  for  the 
transformation  of  integrals.     These   formulae  correspond  to 


198  VECTOR  ANALYSIS 

those  connected  with  "  integration  by  parts  "  in  ordinary 
calculus.  They  are  obtained  by  integrating  both  sides  of  the 
formulse,  page  161,  for  differentiating. 

First  V  (u  i?)  =  u  V  v  +  v  V  u. 

I    V(uv)'di  =    I    wVv«dr=  I    vVu'dr. 
J  c  Jo  J  c 

/    uVv»dr=[uv']     —  lvVu»di.      (14) 

J  r0       " 


Hence 


The  expression  [u  v~\ 

ro 

represents  the  difference  between  the  value  of  (u  v~)  at  r,  the 
end  of  the  path,  and  the  value  at  r0,  the  beginning  of  the  path. 
If  the  path  be  closed 

f  wV  vdi  =  —    C  vV  u»dr.  (14)' 

J  Q  Jo 

Second        Vx  (wv)  =  %Vxv  +  Vwxv. 
ff  V  x  (uv)»d*=  C  C  wVxvcU+  C  C  Vwxvrfa. 

Hence 

rfvwxv^a=  f  uvdr—  f  C  uV  xvda,     (15) 
J  J  a  J  Q  J  J  a 


or 


rfwVxvda=   fwvc?r—  fTVwxv.^a,  (15)' 

Third    Vx(>Vfl)^ttVxVv  +  VwxVtf. 
But  VxVv  =  0 

Hence  v  x  (u  V  v)  =  V  u  x  V  v, 


8 

Hence 


THE  INTEGRAL   CALCULUS  OF  VECTORS         199 

(wvv)«^a=  cr^ux^v'dB.. 


I    i  V  M  x  V  v  •  da.=  i  uV  v»  dr  =  —  C  vV  u*  dr,    (16) 
J  J  s  Jo  Jo 

Fourth  V  •  (u  v)  =  u  V  •  v  +  V  u  •  v. 

liivt(uv)dv  =  rrruvvdv  +  r  r  v  ^  •  v  <z  v. 

Hence 

III    uV*vdv=[  I    uv  da,—  C  i  i  ^U'vdv,  (17) 
or 

I    i  I  v^»v  dv=  I   I    wv^a—  rrTwVv^v,    (17)' 

Fifth     vVwxv   =  VXV^*v  —  Vw»vxv. 


Hence 


f  f  \?  u  x  v  •  da,  =  —  C  i  C  ^  u*\7  x  v  dv.     (18) 


In  all  these  formulae  which  contain  a  triple  integral  the 
surface  S  is  the  closed  surface  bounding  the  body  throughout 
which  the  integration  is  performed. 

Examples  of  integration  by  parts  like  those  above  can  be 
multiplied  almost  without  limit.  Only  one  more  will  be 
given  here.  It  is  known  as  Green's  Theorem  and  is  perhaps 
the  most  important  of  all.  If  u  and  v  are  any  two  scalar 
functions  of  position, 


200  VECTOR  ANALYSIS 


—  v  V  •  V  w, 
/  /  /V^*V^i?=/  /  IV«(«V*)rf«—  I   /  I  u^'Vvdv, 

=  /J   J  V  •  (v  V  w)  ^  v  —  J   J    TV  V  •  V  w  d  0. 
Hence 

/  /  /V^»Vv^v  =  /   /  w  V?>  •  d  a  —  I    /    I  M  V  •  V  v  d  P, 

=r  A>vw«da—  r  f  fvv-vwdv.   (i9> 

By  subtracting  these  equalities  the  formula  (20) 

I   /   I  (wVVv  —  v^7»^u)dv=  I    I  (u^7  v  —  v  V  w)  •  cZ  a. 


is  obtained.  By  expanding  the  expression  in  terms  of  i,  j,  k 
the  ordinary  form  of  Green's  theorem  may  be  obtained.  A 
further  generalization  due  to  Thomson  (Lord  Kelvin)  is  the 
following  : 

/   /  l«V«*V*<fr=i    I  uw^7V'da>—  I    I    I  u^7»[w^7v\dv, 

=  /    I  vwVU'd*  —  I    I    I  v  V  •  [w  V  w]  c?  v,     (21) 

where  t0  is  a  third  scalar  function  of  position. 

The  element  of  volume  dv  has  nothing  to  do  with  the  scalar 
function  v  in  these  equations  or  in  those  that  go  before.  The 
use  of  v  in  these  two  different  senses  can  hardly  give  rise  to 
any  misunderstanding. 

*  85.]  In  the  preceding  articles  the  scalar  and  vector  func- 
tions which  have  been  subject  to  treatment  have  been  sup- 


THE  INTEGRAL   CALCULUS  OF  VECTORS 


201 


posed  to  be  continuous,  single-valued,  possessing  derivatives 
of  the  first  two  orders  at  every  point  of  space  under  consider- 
ation. When  the  functions  are  discontinuous  or  multiple- 
valued,  or  fail  to  possess  derivatives  of  the  first  two  orders 
in  certain  regions  of  space,  some  caution  must  be  exercised  in 
applying  the  results  obtained. 
Suppose  for  instance 


V  K   = 

The  line  integral 


ar  + 


Introducing  polar  coordinates 


xdy  — 


x*  + 


=*•«« 

***    ' 


*  - 


Form  the  line  integral  from  the  point  (+1,0)  to  the  point 
(—1,  0)  along  two  different  paths.  Let  one  path  be  a  semi- 
circle lying  above  the  X-axis ;  and  the  other,  a  semicircle 
lying  below  that  axis.  The  value  of  the  integral  along  the 
first  path  is 


r 


along  the  second  path, 


i  r~" 

-  I   d6  =  ir. 
fj 


From  this  it  appears  that  the  integral  does  not  depend  merely 
upon  the  limits  of  integration,  but  upon  the  path  chosen, 


2Q2  VECTOR  ANALYSIS 

the  value  along  one  path  being  the  negative  of  the  value 
along  the  other.  The  integral  around  the  circle  which  is  a 
closed  curve  does  not  vanish,  but  is  equal  to  ±  2  TT. 

It  might  seem  therefore  the  results  of  Art.  79  were  false 
and  that  consequently  the  entire  bottom  of  the  work  which 
follows  fell  out.  This  however  is  not  so.  The  difficulty  is 

that  the  function 

-i  y 

F=tan     y- 
x 

is  not  single-valued.  At  the  point  (!,!)»  f°r  instance,  the 
function  V  takes  on  not  only  the  value 


F=  tan    1  =  ^» 
4 


but  a  whole  series  of  values 


where  k  is  any  positive  or  negative  integer.  Furthermore  at 
the  origin,  which  was  included  between  the  two  semicircular 
paths  of  integration,  the  function  V  becomes  wholly  inde- 
terminate and  fails  to  possess  a  derivative.  It  will  be  seen 
therefore  that  the  origin  is  a  peculiar  or  singular  point  of  the 
function  V.  If  the  two  paths  of  integration  from  (4-  1,  0)  to 
(—1,0)  had  not  included  the  origin  the  values  of  the  integral 
would  not  have  differed.  In  other  words  the  value  of  the 
integral  around  a  closed  curve  which  does  not  include  the 
origin  vanishes  as  it  should. 

Inasmuch  as  the  origin  appears  to  be  the  point  which 
vitiates  the  results  obtained,  let  it  be  considered  as  marked 
by  an  impassable  barrier.  Any  closed  curve  C  which  does 
not  contain  the  origin  may  be  shrunk  up  or  expanded  at  will  ; 
but  a  closed  curve  C  which  surrounds  the  origin  cannot  be 
so  distorted  as  no  longer  to  enclose  that  point  without  break- 
ing its  continuity.  The  curve  C  not  surrounding  the  origin 


THE  INTEGRAL  CALCULUS  OF  VECTORS  203 

may  shrink  up  to  nothing  without  a  break  in  its  continuity ; 
but  C  can  only  shrink  down  and  fit  closer  and  closer  about 
the  origin.  It  cannot  be  shrunk  down  to  nothing.  It  must 
always  remain  encircling  the  origin.  The  curve  C  is  said  to 
be  reducible  ;  C,  irreducible.  In  case  of  the  function  F",  then, 
it  is  true  that  the  integral  taken  around  any  reducible  circuit 
C  vanishes ;  but  the  integral  around  any  irreducible  circuit  G 
does  not  vanish. 

Suppose  next  that  V  is  any  function  whatsoever.  Let  all 
the  points  at  which  V  fails  to  be  continuous  or  to  have  con- 
tinuous first  partial  derivatives  be  marked  as  impassable 
barriers.  Then  any  circuit  C  which  contains  within  it  no 
such  point  may  be  shrunk  up  to  nothing  and  is  said  to  be 
reducible;  but  a  circuit  which  contains  one  or  more  such 
points  cannot  be  so  shrunk  up  without  breaking  its  continuity 
and  it  is  said  to  be  irreducible.  The  theorem  may  then  be 
stated:  The  line  integral  of  the  derivative  W  of  any  function 
V  vanishes  around  any  reducible  circuit  0.  It  may  or  may  not 
vanish  around  an  irreducible  circuit  In  case  one  irreducible 
circuit  C  may  be  distorted  so  as  to  coincide  with  another 
irreducible  circuit  C  without  passing  through  any  of  the 
singular  points  of  V  and  without  breaking  its  continuity, 
the  two  circuits  are  said  to  be  reconcilable  and  the  values  of 
the  line  integral  of  V  V  about  them  are  the  same. 

A  region  such  that  any  closed  curve  C  within  it  may  be 
shrunk  up  to  nothing  without  passing  through  any  singular 
point  of  V  and  without  breaking  its  continuity,  that  is,  a 
region  every  closed  curve  in  which  is  reducible,  is  said  to  be 
acyclic.  All  other  regions  are  cyclic. 

By  means  of  a  simple  device  any  cyclic  region  may  be  ren- 
dered acyclic.  Consider,  for  instance,  the  region  (Fig.  34)  en- 
closed between  the  surface  of  a  cylinder  and  the  surface  of  a 
cube  which  contains  the  cylinder  and  whose  bases  coincide 
with  those  of  the  cylinder.  Such  a  region  is  realized  in  a  room 


204 


VECTOR  ANALYSIS 


FIG.  84. 


in  which  a  column  reaches  from  the  floor  to  the  ceiling.  It 
is  evident  that  this  region  is  cyclic.  A  circuit  which  passes 
around  the  column  is  irreducible.  It  cannot  be  contracted  to 
nothing  without  breaking  its  continuity.  If 
x — -v  /•  now  a  diaphragm  be  inserted  reaching  from 
"^^  the  surface  of  the  cylinder  or  column  to  the 
surface  of  the  cube  the  region  thus  formed 
bounded  by  the  surface  of  the  cylinder,  the 
surface  of  the  cube,  and  the  two  sides  of  the 
diaphragm  is  acyclic.  Owing  to  the  inser- 
tion of  the  diaphragm  it  is  no  longer  possible 
to  draw  a  circuit  which  shall  pass  completely  around  the  cyl- 
inder —  the  diaphragm  prevents  it.  Hence  every  closed  cir- 
cuit which  may  be  drawn  in  the  region  is  reducible  and  the 
region  is  acyclic. 

In  like  manner  any  region  may  be  rendered  acyclic  by 
inserting  a  sufficient  number  of  diaphragms.  The  bounding 
surfaces  of  the  new  region  consist  of  the  bounding  surfaces  of 
the  given  cyclic  region  and  the  two  faces  of  each  diaphragm. 

In  acyclic  regions  or  regions  rendered  acyclic  by  the  fore- 
going device  all  the  results  contained  in  Arts.  79  et  seq. 
hold  true.  For  cyclic  regions  they  may  or  may  not  hold 
true.  To  enter  further  into  these  questions  at  this  point  is 
unnecessary.  Indeed,  even  as  much  discussion  as  has  been 
given  them  already  may  be  superfluous.  For  they  are  ques- 
tions which  do  not  concern  vector  methods  any  more  than  the 
corresponding  Cartesian  ones.  They  belong  properly  to  the 
subject  of  integration  itself,  rather  than  to  the  particular 
notation  which  may  be  employed  in  connection  with  it  and 
which  is  the  primary  object  of  exposition  here.  In  this 
respect  these  questions  are  similar  to  questions  of  rigor. 


THE  INTEGRAL   CALCULUS  OF  VECTORS         205 

The  Integrating  Operators.     The  Potential 

86.]  Hitherto  there  have  been  considered  line,  surface, 
and  volume  integrals  of  functions  both  scalar  and  vector. 
There  exist,  however,  certain  special  volume  integrals  which, 
owing  to  their  intimate  connection  with  the  differentiating 
operators  V,  V»,  Vx,  and  owing  to  their  especially  frequent 
occurrence  and  great  importance  in  physics,  merit  especial 
consideration.  Suppose  that 


is  a  scalar  function  of  the  position  in  space  of  the  point 


For  the  sake  of  defmiteness  V  may  be  regarded  as  the 
density  of  matter  at  the  point  (x  2,  yv  22).  In  a  homogeneous 
body  V  is  constant.  In  those  portions  of  space  in  which  no 
matter  exists  V  is  identically  zero.  In  non-homogeneous  dis- 
tributions of  matter  V  varies  from  point  to  point;  but  at 
each  point  it  has  a  definite  value. 

The  vector 


drawn  from  any  assumed  origin,  may  be  used  to  designate 
the  point  (#2,  yv  z2).    Let 

On  Vi>  *i) 
be  any  other  fixed  point  of  space,  represented  by  the  vector 

T1  =  x1i  +  y1j  +  z1k 
drawn  from  the  same  origin.     Then 

ra  ~  ri  =  Oa  -  *i)  i  +  Oa  -  yi)  J  +  Oa  -  *i)  k 
is  the  vector  drawn  from  the  point  (xv  yv  «j)  to  the  point 
0&2>  Vv  i?a)<    As  this  vector  occurs  a  large  number  of  times 
in  the  sections  immediately  following,  it  will  be  denoted  by 


12 


206  VECTOR  ANALYSIS 

The  length  of  r12  is  then  r12  and  will  be  assumed  to  be 
positive. 


12  =  V  ria  •  r12  =  V  O2  - 
Consider  the  triple  integral 


The  integration  is  performed  with  respect  to  the  variables 
xv  y?flzi  —  tt*3^  k*  with  respect  to  the   body  of  which   V 
represents  the  density  (Fig.  35).     During 
the   integration  the  point  (xv  yv  z^)  re- 
mains fixed.    The  integral  /  has  a  definite 
j,^      x  ,      value  at  each  definite  point  (xv  yv  Zj). 

It  is  a  function  of  that  point.     The  in- 
FIG.  35. 

terpretation  of  this  integral  /  is  easy,  if 

the  function  V  be  regarded  as  the  density  of  matter  in  space. 
The  element  of  mass  dm  at  (xv  yv  «2)  is 

dm  =  V  (xv  yv  z2)  dxz  dy^  dz%  =  Vdv. 

The  integral  /  is  therefore  the  sum  of  the  elements  of  mass 
in  a  body,  each  divided  by  its  distance  from  a  fixed  point 

VII   3 19   ^1  J  * 

dm 


r    m 
J    r* 


This  is  what  is  termed  the  potential  at  the  point  (xv  yv 
due  to  the  body  whose  density  is 


The  limits  of  integration  in  the  integral  I  may  be  looked  at 
in  either  of  two  ways.  In  the  first  place  they  may  be 
regarded  as  coincident  with  the  limits  of  the  body  of  which 
V  is  the  density.  This  indeed  might  seem  the  most  natural 
set  of  limits.  On  the  other  hand  the  integral  /  may  be 


THE  INTEGRAL   CALCULUS   OF   VECTORS          207 

regarded  as  taken  over  all  space.  The  value  of  the  integral 
is  the  same  in  both  cases.  For  when  the  limits  are  infinite 
the  function  V  vanishes  identically  at  every  point  (#2,  yv  z2) 
situated  outside  of  the  body  and  hence  does  not  augment 
the  value  of  the  integral  at  all.  It  is  found  most  convenient 
to  consider  the  limits  as  infinite  and  the  integral  as  extended 
over  all  space.  This  saves  the  trouble  of  writing  in  special 
limits  for  each  particular  case.  The  function  Fof  itself  then 
practically  determines  the  limits  owing  to  its  vanishing  iden- 
tically at  all  points  unoccupied  by  matter. 

87.]  The  operation  of  finding  the  potential  is  of  such 
frequent  occurrence  that  a  special  symbol,  Pot,  is  used  for  it. 

Pot  V=  ill  d&z  ^y*i  d%y     (22) 

JJJ  fla 

The  symbol  is  read  "the  potential  of  V"  The  potential, 
Pot  V,  is  a  function  not  of  the  variables  xv  yv  z2  with 
regard  to  which  the  integration  is  performed  but  of  the  point 
(*!»  y\)  2i)  which  is  fixed  during  the  integration.  These 
variables  enter  in  the  expression  for  r12.  The  function  V 
and  Pot  V  therefore  have  different  sets  of  variables. 

It  may  be  necessary  to  note  that  although  V  has  hitherto 
been  regarded  as  the  density  of  matter  in  space,  such  an 
interpretation  for  V  is  entirely  too  restricted  for  convenience. 
Whenever  it  becomes  necessary  to  form  the  integral 


of  any  scalar  function  V,  no  matter  what  V  represents,  that 
integral  is  called  the  potential  of  V.  The  reason  for  calling 
such  an  integral  the  potential  even  in  cases  in  which  it  has 
no  connection  with  physical  potential  is  that  it  is  formed 
according  to  the  same  formal  law  as  the  true  potential  and 


208  VECTOR  ANALYSIS 

by  virtue  of  that  formation  has  certain  simple  rules  of  opera- 
tion which  other  types  of  integrals  do  not  possess. 

Pursuant  to  this  idea  the  potential  of  a  vector  function 

Wfrf.          a,         *,     \ 
\*"S>  yy  *27 

may  be  written  down. 

•»  /»W  (Xn,   2/o,    Za) 

dx%  dyz  dzv     (23) 


In  this  case  the  integral  is  the  sum  of  vector  quantities 
and  is  consequently  itself  a  vector.  Thus  the  potential  of  a 
vector  function  W  is  a  vector  function,  just  as  the  potential 
of  a  scalar  function  F'was  seen  to  be  a  scalar  function  of  posi- 
tion in  space.  If  W  be  resolved  into  its  three  components 

W  (z2,  yv  «2)  =  i  X  (#2,  yv  «2)  +  j  Y  (xv  yv  «2) 

+  k  Z  <>2,  yv  z2) 

Pot  W  =  i  PotX  +  j  Pot  Y+  k  Pot  Z.       (24) 

The  potential  of  a  vector  function  W  is  equal  to  the  vector 
sum  of  the  potentials  of  its  three  components  X,  I7",  Z. 

The  potential  of  a  scalar  function  V  exists  at  a  point 
(#i»  y\t  *i>)  when  and  only  when  the  integral 


taken  over  all  space  converges  to  a  definite  value.  If, 
for  instance,  V  were  everywhere  constant  in  space  the  in- 
tegral would  become  greater  and  greater  without  limit  as 
the  limits  of  integration  were  extended  farther  and  farther 
out  into  space.  Evidently  therefore  if  the  potential  is  to  exist 
V  must  approach  zero  as  its  limit  as  the  point  (#2,  yv  za) 
recedes  indefinitely.  A  few  important  sufficient  conditions 
for  the  convergence  of  the  potential  may  be  obtained  by 
transforming  to  polar  cob'rdinates.  Let 


THE  INTEGRAL   CALCULUS  OF  VECTORS          209 

x  =  r  sin  6  cos  <£, 
y  —  r  sin  0  sin  <f>, 

z  =  r  cos  0, 
d  v  =  r2  sin  0  rf  r  d  0  d  <£. 

Let  the  point  (a^,  yv  2j)  which  is  fixed  for  the  integration 
be  chosen  at  the  origin.     Then 

ri2  =  r 
and  the  integral  becomes 


fff       dV*  =  fffT*  aw0  dr  d0  dfr  (22) 

or  simply          PotF=   CCfVrsmO  dr  dO  d<j>. 

If  the  function  V  decrease  so  rapidly  that  the  product 


remains  finite  as  r  increases  indefinitely,  then  the  integral  con- 
verges as  far  as  the  distant  regions  of  space  are  concerned. 
For  let 


r  =  00 


me  dr  d0 


=R 


=  GO 


8m0  dr  d0 


r-R 


r  =  oo 


Hence  the  triple  integral  taken  over  all  space  outside  of  a 
sphere  of  radius  R  (where  R  is  supposed  to  be  a  large  quan- 
tity) is  less  than  4  TT  K  /R,  and  consequently  converges  as  far 
as  regions  distant  from  the  origin  are  concerned. 


210  VECTOR  ANALYSIS 

If  the  function  V  remain  finite  or  if  it  become  infinite  so 

weakly  that  the  product 

Vr 

remains  finite  when  r  approaches  zero,  then  the  integral  converges 
as  far  as  regions  near  to  the  origin  are  concerned.    For  let 

Vr<K 

r=R  r=R 

III  ^r  s*n  & dr  d  0  d<f>    <    I    I    I  Kd r  dd  d<j>. 

r  =  Q  r=0 

r  =  R 

Kdr  dO  dd>  = 


C  C  C 


Hence  the  triple  integral  taken  over  all  space  inside  a  sphere 
of  radius  R  (where  E  is  now  supposed  to  be  a  small  quantity) 
is  less  than  4  TT  K  E  and  consequently  converges  as  far  as 
regions  near  to  the  origin  which  is  the  point  (xv  yv  zj  are 
concerned. 

If  at  any  point  (x2,  y2,  z2)  not  coincident  with  the  origin, 
i.  e.  the  point  (x  v  yx,  z  x),  the  function  V  becomes  infinite  so 
weakly  that  the  product  of  the  value  0/V  at  a  point  near  to 
(x2,  y2,  z2)  by  the  square  of  the  distance  of  that  point  from 
(x2,  y2,  z2)  remains  Jinite  as  that  distance  approaches  zero,  then 
the  integral  converges  as  far  as  regions  near  to  the  point  (x2,  y2,  z2) 
are  concerned.  The  proof  of  this  statement  is  like  those  given 
before.  These  three  conditions  for  the  convergence  of  the 
integral  Pot  V  are  sufficient.  They  are  by  no  means  neces- 
sary. The  integral  may  converge  when  they  do  not  hold. 
It  is  however  indispensable  to  know  whether  or  not  an  integral 
under  discussion  converges.  Unless  the  tests  given  above 
show  the  convergence,  more  stringent  ones  must  be  resorted 
to.  Such,  however,  will  not  be  discussed  here.  They  belong 
to  the  theory  of  integration  in  general  rather  than  to  the 


THE  INTEGRAL   CALCULUS  OF   VECTORS 


211 


theory  of  the  integrating  operator  Pot.  The  discussion  of 
the  convergence  of  the  potential  of  a  vector  function  W  re- 
duces at  once  to  that  of  its  three  components  which  are  scalar 
functions  and  may  be  treated  as  above. 

88.]  The  potential  is  a  function  of  the  variables  xv  y^  z1 
which  are  constant  with  respect  to  the  integration.  Let  the 
value  of  the  potential  at  the  point  (xv  yv  zj  be  denoted  by 


The  first  partial  derivative  of  the  potential  with  respect  to  x1 
is  therefore 


The  value  of  this  limit  may  be  determined  by  a  simple 
device    (Fig.   36).      Consider 
the  potential  at  the  point 

due  to  a  certain  body  T.  This 
is  the  same  as  the  potential  at 
the  point 

due  to  the  same  body  T  displaced  in  the  negative  direction  by 
the  amount  A  xr  For  in  finding  the  potential  at  a  point  P 
due  to  a  body  T  the  absolute  positions  in  space  of  the  body 
T  and  the  point  P  are  immaterial.  It  is  only  their  positions 
relative  to  each  other  which  determines  the  value  of  the  poten- 
tial. If  both  body  and  point  be  translated  by  the  same 
amount  in  the  same  direction  the  value  of  the  potential  is  un- 
changed. But  now  if  T  be  displaced  in  the  negative  direction 
by  the  amount  A  #,  the  value  of  V  at  each  point  of  space  is 
changed  from 


to 


where  A  xz  =  A  xr 


212  VECTOR  ANALYSIS 

Hence 

[Pot  FOa,^,^)]^  +  A*,  ,M  *  =  [Pot  FOa 

Hence         LlM      $  [Pot  H****.*.*-  [Pot 


^  )  = 
) 


It  will  be  found  convenient  to  introduce  the  limits  of 
integration.  Let  the  portion  of  space  originally  filled  by  the 
body  T  be  denoted  by  M  ;  and  let  the  portion  filled  by  the 
body  after  its  translation  in  the  negative  direction  through 
the  distance  A  xl  be  denoted  by  M'.  The  regions  M  and  M' 
overlap.  Let  the  region  common  to  both  be  M  ;  and  let  the 
remainder  of  M  be  m;  the  remainder  of  M'  ,  mr.  Then 


/»/*/*     V  f  r  -4-  A  v 

_  err  *  wtffj 

^vzz)  j          CCC    ^(^a+Aa^y^)  j 

J  J  J  M             r12 

-  r  r  r  r(^  y2' 

rt  va  1    1  I  1                                *  «r 

JJJm'                 T12 

f*     f*     f*       V  f  V         11        9    \ 

)l     1      1             v    2»  "a'     3x   ^7  ,,, 

—  If                                              "  V2 

JJJM          r12 

»  "\                      f*   f*   f*     T^  (  Y      ni      9.   ^\ 
^  dr     1     1     1     1          ^    2'  ^2'     2^  (fr 

J  J  J  H          r12 

Hence  (25)  becomes, 

ava+                                                av2. 
JJJm           r  12 

when  A  a;1  is  replaced  by  its  equal  A  #2, 

t  As  all  the  following  potentials  are  for  the  point  xv  yit  z^  the  bracket  and 
indices  have  been  dropped. 


THE  INTEGRAL   CALCULUS   OF  VECTORS          213 

f       (*(*(*(  VT     -I- A  f       II      9    \  f*  f*  f*     T^V/n       ni 

(  l^2+Ag2,y2,z2;       _  rrr  y^vy 

LlM    <  JJJfi Tja J  J  J  M        ru 

Aa;2=o(  Az2  =  0 


a 

la 

or, 


12 


A 

A  a;  = 


LIM    CCC 

A*=0t/  «/  J  m 


LIM  rrr 

X±QjJJx 


+  A  g  y  *>  -  ^c^a^  y^  »»>  , 


9 


when  A  Xj  approaches  zero  as  its  limit  the  regions  mand  TO', 
which  are  at  no  point  thicker  than  A  a,  approach  zero  ;  M  ' 
and  M  both  approach  M  as  a  limit. 

t  There  are  cases  in  which  this  reversal  of  the  order  in  which  the  two  limits 
are  taken  gives  incorrect  results.  This  is  a  question  of  double  limits  and  leads  to 
the  mazes  of  modern  mathematical  rigor. 

}  If  the  derivative  of  Fis  to  exist  at  the  surface  bounding  T  the  values  of  the 
function  V  must  diminish  continuously  to  zero  upon  the  surface.  If  Fchanged 
suddenly  from  a  finite  value  within  the  surface  to  a  zero  value  outside  the  de- 
rivative 5  VI  3  *i  would  not  exist  and  the  triple  integral  would  be  meaningless. 
For  the  same  reason  V  is  supposed  to  be  finite  and  continuous  at  every  point 
within  the  region  T. 


214  VECTOR  ANALYSIS 

Then  if  it  be  assumed  that  the  region  Tis  finite  and  that  V 
vanishes  upon  the  surface  bounding  T 


5.//L 


LlM         111         r    v^fl   i   •-•  *"a»  #2'  ~ay    7  „ 

V«  =  v 


12 


A»2  =  ' 

Consequently  the  expression  for  the  derivative  of  the  poten- 
tial reduces  to  merely 


.   (26) 


5%e  partial  derivative  of  the  potential  of  a  scalar  function  V 
is  equal  to  the  potential  of  the  partial  derivative  of  V. 

The  derivative  V  of  the  potential  of^fis  equal  to  the  potential 
of  the  derivative  V  V. 

VPotF=PotVr:  (27) 

This  statement  follows  immediately  from  the  former.  As 
the  V  upon  the  left-hand  side  applies  to  the  set  of  vari- 
ables xv  yv  2j,  it  may  be  written  Vr  In  like  manner  the 
V  upon  the  right-hand  side  may  be  written  V2  to  call  atten- 
tion to  the  fact  that  it  applies  to  the  variables  scv  yv  zz  of  V. 

Then  Vj  PotF=  Pot  V2F!  (27)' 

To  demonstrate  this  identity  V  may  be  expanded  in  terms  of 

.3PotF      .SPotF       t 

i L  i L.   V 

*J 


THE  INTEGRAL   CALCULUS  OF  VECTORS         215 

As  i,  j,  k  are  constant  vectors  they  may  be  placed  under 
the  sign  of  integration  and  the  terms  may  be  collected.  Then 
by  means  of  (26) 


The  curl  V  X  and  divergence  V  •  of  the  potential  of  a  vector 
function  W  are  equal  respectively  to  the  potential  of  the  curl  and 
divergence  of  that  function. 

Vj  x  Pot  W  =  Pot  V2  x  W, 
or  curl  Pot  W  =  Pot  curl  W 

and  Vj  •  Pot  W  =  Pot  V2  •  W, 

or  div  Pot  W  =  Pot  div  W. 

These  relations  may  be  proved  in  a  manner  analogous  to  the 
above.  It  is  even  possible  to  go  further  and  form  the  dels 
of  higher  order 

V  •  V  Pot  F=  Pot  V  •  VF,  (30) 

V  •  V  Pot  W  =  Pot  V  •  V  W,  (31) 

VV  •  Pot  W  =  Pot  V  V  •  W,  (32) 

VxV  x  Pot  W=  Pot  Vx  VxW.  (33) 

The  dels  upon  the  left  might  have  a  subscript  1  attached  to 
show  that  the  differentiations  are  performed  with  respect  to 
the  variables  xv  y±,  zv  and  for  a  similar  reason  the  dels  upon 
the  right  might  have  been  written  with  a  subscript  2.  The 
results  of  this  article  may  be  summed  up  as  follows  : 

Theorem:  The  differentiating  operator  V  and  the  integrating 
operator  Pot  are  commutative. 

*89.]  In  the  foregoing  work  it  has  been  assumed  that  the 
region  T  was  finite  and  that  the  function  Fwas  everywhere 
finite  and  continuous  inside  of  the  region  T  and  moreover 
decreased  so  as  to  approach  zero  continuously  at  the  surface 
bounding  that  region.  These  restrictions  are  inconvenient 


216  VECTOR  ANALYSIS 

and  may  be  removed  by  making  use  of  a  surface  integral. 
The  derivative  of  the  potential  was  obtained  (page  213)  in 
essentially  the  form 

PPotF 

— 
9 


otF      CCC     1  sv 
=  111     -- 

xl       J  J  J  M  r12*>xz 


Lra      1 


LIM       1 

9 


Let  d  a  be  a  directed  element  of  the  surface  S  bounding  the 
region  M.  The  element  of  volume  dvz  in  the  region  mr  is 
therefore  equal  to 


Hence        _L  f  f  f    ^(' 
A*2J  J  J  m> 

=  r  r  v<jx* 

J  J 


d 


12 


T12 

The  element  of  volume  d  vz  in  the  region  m  is  equal  to 
d  02  =  —  A  xz  i  •  d  a. 

1        /•/»/»   TT'f'v     ti     <*  \ 
ill       \  v  &v    is  j 

rlenco  —  -7 —  I        '    A  Vn 

A  a;,  J  J  J  m          r12 


Consequently 


1 9V  r r    V 

—  -r — dvi+l         — i.rfa.      (34) 
rlzdx^  J  J  a  r12 


THE  INTEGRAL   CALCULUS  OF  VECTORS          217 

The  volume  integral  is  taken  throughout  the  region  M  with 
the  understanding  that  the  value  of  the  derivative  of  V  at 
the  surface  S  shall  be  equal  to  the  limit  of  the  value  of  that 
derivative  when  the  surface  is  approached  from  the  interior 
of  M.  This  convention  avoids  the  difficulty  that  arises  in 
connection  with  the  existence  of  the  derivative  at  the  surface 
S  where  V  becomes  discontinuous.  The  surface  integral  is 
taken  over  the  surface  S  which  bounds  the  region. 

Suppose  that  the  region  M  becomes  infinite.  By  virtue  of 
the  conditions  imposed  upon  V  to  insure  the  convergence  of 

the  potential 

<  K. 


Let  the  bounding  surface  S  be  a  sphere  of  radius  -B,  a  quan- 
tity which  is  large. 

i  •  d&  <  r2  dO  d<f>. 


The  surface  integral  becomes  smaller  and  smaller  and  ap- 
proaches zero  as  its  limits  when  the  region  M  becomes  infinite. 
Moreover  the  volume  integral 


—  « —  dv«, 

rev  IP 
•tn     \s  wn 

remains  finite  as  M  becomes  infinite.  Consequently  provided 
V  is  such  a  function  that  Pot  V  exists  as  far  as  the  infinite 
regions  of  space  are  concerned,  then  the  equation 

SPotF  9V 

— ^ =  Pot  — 

holds  as  far  as  those  regions  of  space  are  concerned. 

Suppose  that  V  ceases  to  be  continuous  or  becomes  infinite 
at  a  single  point  (xv  yv  Zj)  within  the  region  T.     Surround 


218  VECTOR  ANALYSIS 

this  point  with  a  small  sphere  of  radius  JR.  Let  S  denote  the 
surface  of  this  sphere  and  M  all  the  region  T  not  included 
within  the  sphere.  Then 

3PotF        r  r  r    1    5F  C  f     V  . 

—  ~  --  =  /   /   /     —  «  —  dv2+    I    I       -i. 

9*i          J  J  J*ru9xs  J  Js   r12 

By  the  conditions  imposed  upon  V 

Vr<K 


f  f  —i.rfa    <    ff  K  dd 


Consequently  when  the  sphere  of  radius  R  becomes  smaller 
and  smaller  the  surface  integral  may  or  may  not  become  zero. 
Moreover  the  volume  integral 


f*  f*  f*     1    PF 

J   J  J M  r!2    5xZ 


may  or  may  not  approach  a  limit  when  R  becomes  smaller 
and  smaller.  Hence  the  equation 

PPotF  5F 

_    —    l-*f\+ 
^  —   J.  \Jv     _ 

dxl  dx% 

has  not  always  a  definite  meaning  at  a  point  of  the  region 
T  at  which  F  becomes  infinite  in  such  a  manner  that  the 
product  Fr  remains  finite. 

If,  however,  F  remains  finite  at  the  point  in  question  so 
that  the  product  Fr  approaches  zero,  the  constant  K  is  zero 
and  the  surface  integral  becomes  smaller  and  smaller  as  R 
approaches  zero.  Moreover  the  volume  integral 


///. 


2 


THE  INTEGRAL   CALCULUS   OF  VECTORS          219 

approaches  a  definite  limit  as  R  becomes  infinitesimal.     Con- 
sequently the  equation 

5PotFPot5T 


holds  in  the  neighborhood  of  all  isolated  points  at  which  V 
remains  finite  even  though  it  be  discontinuous. 

Suppose  that  V  becomes  infinite  at  some  single  point 
(«2,  y2,  z2)  not  coincident  with  (a^,  y±,  Zj).  According  to  the 
conditions  laid  upon  V 

VI*  <  K, 

where  I  is  the  distance  of  the  point  (xv  yv  «2)  from  a  point 
near  to  it.     Then  the  surface  integral 

V  . 

r!2 

need  not  become  zero  and  consequently  the  equation 

PPotF  5V 

— ^ =  Pot  •= — 

O  #j  C'S'a 

need  not  hold  for  any  point  (x^  yv  z^)  of  the  region.     But 
if  V  becomes  infinite  at  #2,  yv  z2  in  such  a  manner  that 

VI  <K, 

then  the  surface  integral  will  approach  zero  as  its  limit  and 
the  equation  will  hold. 

Finally  suppose  the  function  V  remains  finite  upon  the 
surface  S  bounding  the  region  jP,  but  does  not  vanish  there. 
In  this  case  there  exists  a  surface  of  discontinuities  of  V. 
Within  this  surface  V  is  finite ;  without,  it  is  zero.  The 
surface  integral 

—  i»  da 


220  VECTOR  ANALYSIS 

does  not  vanish  in  general.     Hence  the  equation 
PPotF  9V 

— jr =   Pot    5 

d  xl  dx% 

cannot  hold. 

Similar  reasoning  may  be  applied  to  each  of  the  three 
partial  derivatives  with  respect  to  xv  yv  zr  By  combining 
the  results  it  is  seen  that  in  general 

Vj  PotF=  Pot  V2F  +    ff    J-  d&.        (35) 

Let  V  be  any  function  in  space,  and  let  it  be  granted  that 
Pot  F  exists.  Surround  each  point  of  space  at  which  V 
ceases  to  be  finite  by  a  small  sphere.  Let  the  surface  of  the 
sphere  be  denoted  by  S.  Draw  in  space  all  those  surfaces 
which  are  surfaces  of  discontinuity  of  V.  Let  these  sur- 
faces also  be  denoted  by  8.  Then  the  formula  (35)  holds 
where  the  surface  integral  is  taken  over  all  the  surfaces 
which  have  been  designated  by  S.  If  the  integral  taken 
over  all  these  surfaces  vanishes  when  the  radii  of  the  spheres 
above  mentioned  become  infinitesimal,  then 

V1PotF=PotV2F  (27)' 

This  formula 

V1PotF=PotV2F. 

will  surely  hold  at  a  point  (x1?  yx,  zx)  if  V  remains  always 
finite  or  becomes  infinite  at  a  point  (x^  y2,  z3)  so  that  the 
product  V 1  remains  finite,  and  if  V  possesses  no  surfaces  of 
discontinuity ,  and  if  furthermore  the  product  V  r3  remains  finite 
as  r  becomes  infinite.1  In  other  cases  special  tests  must  be 
applied  to  ascertain  whether  the  formula  (27) '  can  be  used 
or  the  more  complicated  one  (35)  must  be  resorted  to. 

1  For  extensions  and  modifications  of  this  theorem,  see  exercises. 


THE  INTEGRAL   CALCULUS  OF  VECTORS          221 

The  relation  (27)  is  so  simple  and  so  amenable  to  trans- 
formation that  V  will  in  general  be  assumed  to  be  such  a 
function  that  (27)  holds.  In  cases  in  which  V  possesses  a 
surface  S  of  discontinuity  it  is  frequently  found  convenient 
to  consider  V  as  replaced  by  another  function  V  which  has 
in  general  the  same  values  as  F'but  which  instead  of  possess- 
ing a  discontinuity  at  S  merely  changes  very  rapidly  from 
one  value  to  another  as  the  point  (#2,  yv  22)  passes  from  one 
side  of  S  to  the  other.  Such  a  device  renders  the  potential 
of  V  simpler  to  treat  analytically  and  probably  conforms  to 
actual  physical  states  more  closely  than  the  more  exact 
conception  of  a  surface  of  discontinuity.  This  device  prac- 
tically amounts  to  including  the  surface  integral  in  the 
symbol  Pot  V  V. 

In  fact  from  the  standpoint  of  pure  mathematics  it  is 
better  to  state  that  where  there  exist  surfaces  at  which  the 
function  V  becomes  discontinuous,  the  full  value  of  Pot  V  V 
should  always  be  understood  as  including  the  surface  integral 


a  '  12 

in  addition  to  the  volume  integral 

VF 


fff 


12 

In  like  manner  Pot  V  •  W,  Pot  V  X  W,  New  V  •  W  and  other 
similar  expressions  to  be  met  in  the  future  must  be  regarded 
as  consisting  not  only  of  a  volume  integral  but  of  a  surface 
integral  in  addition,  whenever  the  vector  function  W  possesses 
a  surface  of  discontinuities. 

It  is  precisely  this  convention  in  the  interpretation  of 
formulse  which  permits  such  simple  formulae  as  (27)  to  hold 
in  general,  and  which  gives  to  the  treatment  of  the  integrat- 
ing operators  an  elegance  of  treatment  otherwise  unobtainable. 


222  VECTOR  ANALYSIS 

The  irregularities  which  may  arise  are  thrown  into  the  inter- 
pretation, not  into  the  analytic  appearance  of  the  formulae. 
This  is  the  essence  of  Professor  Gibbs's  method  of  treatment. 
90.]  The  first  partial  derivatives  of  the  potential  may  also 
be  obtained  by  differentiating  under  the  sign  of  integration.1 


"  V) 


gotr  *-  *- 

2 


In  like  manner  for  a  vector  function  W 

' 


~  c>Potr 

\jr  — « : 

&X1 

and  — =  III  — — g— ^ —  dvv  (38)' 

1  12 

V  Pot  F=  i     ^^H-  J     n0^  +  k 


But        i  (a?a  -ajj)  +  j  (ya  -yj)  +  k  («a  -  Zj)  =  r12. 

1  If  an  attempt  were  made  to  obtain  the  second  partial  derivatives  in  the  same 
manner,  it  would  be  seen  that  the  volume  integrals  no  longer  converged. 


THE  INTEGRAL   CALCULUS  OF  VECTORS          223 

Hence  V  Pot  F=  fff  Tjf~  *  *r  (39) 

In  like  manner 

VxPotW=  III   "^"3 dvv  (40) 

and  V.PotW=  f  f  f  ^f—  dv*  (41) 

J  J  J       r  12 

These  three  integrals  obtained  from  the  potential  by  the 
differentiating  operators  are  of  great  importance  in  mathe- 
matical physics.  Each  has  its  own  interpretation.  Conse- 
quently although  obtained  so  simply  from  the  potential  each 
is  given  a  separate  name.  Moreover  inasmuch  as  these 
integrals  may  exist  even  when  the  potential  is  divergent, 
they  must  be  considered  independent  of  it.  They  are  to 
be  looked  upon  as  three  new  integrating  operators  defined 
each  upon  its  own  merits  as  the  potential  was  defined. 

Let,  therefore, 


(42) 


'   12 
r12x  W  (#2>y2>z2)  ^      ,      , 

"„  '  Cv  30n  Cv  £/ n  Cv  %n   ~ 

A«O  m       *r  m  m 

r    12 

— 32      '   2    dxzdyzdzz  =  Ma,x.Vf.     (44) 

r  12 

If  the  potential  exists,  then 

V  Pot  F=  New  F 

V  X  Pot  W  =  Lap  W  (45) 

V  •  Pot  W  =  Max  W. 
The  first  is  written  New  F  and  read  "  The  Newtonian  of  F." 


224  VECTOR  ANALYSIS 

The  reason  for  calling  this  integral  the  Newtonian  is  that  if 
V  represent  the  density  of  a  body  the  integral  gives  the  force 
of  attraction  at  the  point  (xv  yv  z^)  due  to  the  body.  This 
will  be  proved  later.  The  second  is  written  Lap  W  and 
read  "  the  Laplacian  of  W."  This  integral  was  used  to  a 
considerable  extent  by  Laplace.  It  is  of  frequent  occurrence 
in  electricity  and  magnetism.  If  W  represent  the  current 
C  in  space  the  Laplacian  of  C  gives  the  magnetic  force  at  the 
point  (xv  yv  zj  due  to  the  current.  The  third  is  written 
Max  W  and  read  "  the  Maxwellian  of  W."  This  integral  was 
used  by  Maxwell.  It,  too,  occurs  frequently  in  electricity 
and  magnetism.  For  instance  if  W  represent  the  intensity 
of  magnetization  I,  the  Maxwellian  of  I  gives  the  magnetic 
potential  at  the  point  (xv  yv  z-^)  due  to  the  magnetization. 

To  show  that  the  Newtonian  gives  the  force  of  attraction 
according  to  the  law  of  the  inverse  square  of  the  distance. 
Let  dmz  be  any  element  of  mass  situated  at  the  point 
(#2*  y-K>  zz)'  The  force  at  (xv  yv  z^)  due  to  dm  is  equal  to 

d  mn 


12 


in  magnitude  and  has  the  direction  of  the  vector  r12  from  the 
point  (xv  yv  zj  to  the  point  (xv  yv  22).    Hence  the  force  is 


12 


Integrating  over  the  entire  body,  or  over  all  space  according 
to  the  convention  here  adopted,  the  total  force  is 


where  V  denotes  the  density  of  matter. 


THE  INTEGRAL   CALCULUS  OF  VECTORS         225 
The  integral  may  be  expanded  in  terms  of  i,  j,  k, 


12 


The  three  components  may  be  expressed  in  terms  of  the  po- 
tential (if  it  exists)  as 


(42)' 


It  is  in  this  form  that  the  Newtonian  is  generally  found  in 
books. 

To  show  that  the  Laplacian  gives  the  magnetic  force  per 
unit  positive  pole  at  the  point  (xv  yv  2j)  due  to  a  distribution 
W  («2,  2/2,  z2)  of  electric  flux.  The  magnetic  force  at  (xv  yv  zx) 
due  to  an  element  of  current  d  C2  is  equal  in  magnitude  to 
the  magnitude  d  Oz  of  that  element  of  current  divided  by  the 
square  of  the  distance  r12  ;  that  is 


The  direction  of  the  force  is  perpendicular  both  to  the  vector 
element  of  current  dC2  and  to  the  line  r12  joining  the  points. 
The  direction  of  the  force  is  therefore  the  direction  of  the 
vector  product  of  r12  and  dGz.  The  force  is  therefore 


12 
15 


226  VECTOR  ANALYSIS 

Integrating  over  all  space,  the  total  magnetic  force  acting  at 
the  point  (xv  yv  zj  upon  a  unit  positive  pole  is 


.3 
12 


This  integral  may  be  expanded  in  terms  of  i,  j,  k.     Let 
W  O2, 2/2,  za)  =  *  -Xfyv  2/2»  *a)  +  J  Y(xv  2/2'  ^2) 


'12  =  («a  -  «!>  i  +  (2/2  -  «/i)  J  +  (»a  -  * 
The  i,  j,  k  components  of  Lap  W  are  respectively 


(43)' 

(200 


k  •  Lap  W  = 

In  terms  of  the  potential  (if  one  exists)  this  may  be  writt 

Pot  Z     9  Pot  T 


i  •  Lap  W  = 


9zl 


5PotF 

=  -^ = 

9xl  dyl 

To  show  that  if  I  be  the  intensity  of  magnetization  at  the 
point  (xv  ^2,  z 2),  that  is,  if  I  be  a  vector  whose  magnitude  is 
equal  to  the  magnetic  moment  per  unit  volume  and  whose 


THE  INTEGRAL   CALCULUS  OF  VECTORS          227 

direction  is  the  direction  of  magnetization  of  the  element  d  v2 
from  south  pole  to  north  pole,  then  the  Maxwellian  of  I  is  the 
magnetic  potential  due  to  the  distribution  of  magnetization. 
The  magnetic  moment  of  the  element  of  volume  d  v%  is  I  d  vz. 
The  potential  at  (%vyv  Zj)  due  to  this  element  is  equal  to  its 
magnetic  moment  divided  by  the  square  of  the  distance  r12 
and  multiplied  by  the  cosine  of  the  angle  between  the  direc- 
tion of  magnetization  I  and  the  vector  rla.  The  potential  is 
therefore 


r  12 

Integrating,  the  total  magnetic  potential  is  seen  to  be 

'i2'1 ->?,,,  =  Max  I. 


•*> 

12 


This  integral  may  also  be  written  out  in  terms  of  x,  y,  z. 
Let 


JA  +  (2/2  - 

If  instead  of  xv  yv  z1  the  variables  x,  y,  z;  and  instead  of 
xv  Vy  22  tne  variables  f,  ?/,  ^  be  used  l  the  expression  takes 
oil  the  form  given  by  Maxwell. 

Max  I  =  ///  {<a<f-»)  +  i(f-f)+  C(S-z)  \  ^dv. 
According  to  the  notation  employed  for  the  Laplacian 
W  -  f  f  f  (X*  ~ 


Max  *       *  a  ~  y'}  Y+  fa  ~  '- 


(44)' 


Maxwell  :  Electricity  and  Magnetism,  Vol.  II.  p.  9. 


228  VECTOR  ANALYSIS 

The  Maxwellian  of  a  vector  function    is  a  scalar  quantity. 
It  may  be  written  in  terms  of  the  potential  (if  it  exists)  as 


c>Potr 

Max  W  =  —  =  --  +  —5-    -  +  —  =  ---       (44)" 


This  form  of  expression  is  much  used  in  ordinary  treatises 
upon  mathematical  physics. 

The  Newtonian,  Laplacian,  and  Maxwellian,  however,  should 
not  be  associated  indissolubly  with  the  particular  physical 
interpretations  given  to  them  above.  They  should  be  looked 
upon  as  integrating  operators  which  may  be  applied,  as  the 
potential  is,  to  any  functions  of  position  in  space.  The  New- 
tonian is  applied  to  a  scalar  function  and  yields  a  vector 
function.  The  Laplacian  is  applied  to  a  vector  function 
and  yields  a  function  of  the  same  sort.  The  Maxwellian 
is  applied  to  a  vector  function  and  yields  a  scalar  function. 
Moreover,  these  integrals  should  not  be  looked  upon  as  the 
derivatives  of  the  potential.  If  the  potential  exists  they 
are  its  derivatives.  But  they  frequently  exist  when  the 
potential  fails  to  converge. 

9L]  Let  V  and  W  be  such  functions  that  their  potentials 
exist  and  have  in  general  definite  values.  Then  by  (27)  and 
(29) 

V«VPotF=V.PotVF=Pot  V.VF. 

But  by  (45)  V  Pot  V  =  New  F, 

and  V.Pot  VF=Max  W. 

Hence         V.  V  PotF  =  V  •  NewF  =  Max  VF 

=  Pot  V.VF    (46) 

By  (27)  and  (29)  VV  .Pot  W  =  VPot  V.  W  =  Pot  V  V.  W. 
But  by  (45)  V  •  Pot  W  =  Max  W, 

and  by  (45)         V  Pot  V  •  W  =  New  V  •  W. 


THE  INTEGRAL   CALCULUS  OF   VECTORS          229 
Hence  V  V  •  Pot  W  =  V  Max  W  =  New  V  W 

=  Pot  VV •  W      (47) 

By  (28)         V  x  V  x  Pot  W  =  V  x  Pot  V  x  W 

=  Pot  V  x  V  x  W. 

But  by  (45)  V  x  Pot  W  =  Lap  W, 

and  V  x  Pot  V  x  W  =  Lap  V  x  W. 

Hence        V  x  V  x  Pot  W  =  V  x  Lap  W  =  Lap  V  x  W 

=  Pot  V  x  V  x  W.  (48) 

By  (56),  Chap.  III.        V .  V  x  Pot  W  =  0, 
or  V  •  Pot  V  x  W  =  0. 

Hence  V  •  Lap  W  =  Max  V  x  W  =  0.  (49) 

And  by  (52),  Chap.  III.     V  x  V  PotF=  0, 
or  VxPotVF=0. 

Hence  V  x  New  V  =  Lap  V  V  =  0.  (50) 

And  by  (58),  Chap.  III.     V  x  V  x  W  =  VV-  W-  V-  VW, 

V.VW  =  VV.W-VxVxW. 

Hence         V  •  V  W  =  New  V  •  W  -  Lap  V  x  W,         (51) 
or  V  •  V  W  =  V  Max  W  —  V  x  Lap  W. 

These  formulae  may  be  written  out  in  terms  of  curl  and 
div  if  desired.     Thus 

div  New  V  =  Max  V  F,  (46)' 

V  Max  W  =  New  div  W  (47)' 

curl  Lap  W  =  Lap  curl  W  (48)' 

div  Lap  W  =  Max  curl  W  =  0  (49)' 

curl  New  V  =  Lap  V  V  =  0  (50)' 

V  •  V  W  =  New  div  W  -  Lap  curl  W.  (51)' 


230  VECTOR  ANALYSIS 

Poisson's  Equation 
92.]    Let  V  "be  any  function  in  space  such  that  the  potential 

PotF 
has  in  general  a  definite  value.     Then 

V.VPotF=-47rF,  (52) 

c>2PotF      P2PotF      22PotF 

or          •    0     a     +      02      +      02     =  -47rF. 
dx{  dy-f  9  9f 

This  equation  is  known  as  Poisson's  Equation. 

The  integral  which  has  been  defined  as  the  potential  is  a 
solution  of  Poisson's  Equation.     The  proof  is  as  follows. 


j  Pot  F=  Vj  •  New  F=  Max  V2  ^= 


12 


The  subscripts  1  and  #  have  been  attached  to  designate 
clearly  what  are  variables  with  respect  to  which  the  differen- 
tiations are  performed. 


=  Vj  •  NewF=  C  C  C^7l  --  •  V2 


But  Vj  —  =  -  V2 


and  V2.     F  V2  =V2  1-  •  V2F+  F  V2  .  V2 


THE  INTEGRAL   CALCULUS  OF   VECTORS          231 
Hence    -  V2  —  .  V2F  =  T V2  .  V2  i  - V2/F  V2  J 

1  1 

or       Vt  —  •  v   K  =  K  v    •  V I- V 

1  r  2  22  2 

'12  r  12 

Integrate : 


+ 


But  V2  •  V2  —  =  0. 

*H 

That  is  to  say  —  satisfies  Laplace's  Equation.    And  by  (8) 


Hence        Vx  •  Vx  Pot  F=  T  T  TVj  --  •  V2  T<£  v2     (53) 

=  f  f  F  YI  —•<*«. 

J  J  j  r12 

The  surface  integral  is  taken  over  the  surface  which  bounds 
the  region  of  integration  of  the  volume  integral.  This  is 
taken  "  over  all  space."  Hence  the  surface  integral  must  be 
taken  over  a  sphere  of  radius  R,  a  large  quantity,  and  R  must 
be  allowed  to  increase  without  limit.  At  the  point  (xv  y^z^), 
however,  the  integrand  of  the  surface  integral  becomes  in- 
finite owing  to  the  presence  of  the  term 


12 


232  VECTOR  ANALYSIS 

Hence  the  surface  S  must  include  not  only  the  surface  of  the 
sphere  of  radius  R>  but  also  the  surface  of  a  sphere  of  radius 
jK',  a  small  quantity,  surrounding  the  point  (xv  yv  z^)  and  R  ' 
must  be  allowed  to  approach  zero  as  its  limit. 

As  it  has  been  assumed  that  the  potential  of  F"  exists,  it  is 
assumed  that  the  conditions  given  (Art.  87)  for  the  existence 
of  the  potential  hold.  That  is 

F"r3  <  K,  when  r  is  large 
Vr  <  K,  when  r  is  small. 

Introduce  polar  coordinates  with  the  origin  at  the  point 
(xv  yv  Zj).  Then  r12  becomes  simply  r 

and  V   —  =  -V  —  =       * 


Then  for  the  large  sphere  of  radius  R 

1  r 

V,  --.da  =  —  jrrasin0d0  d&. 
r12  r8 

Hence  the  surface  integral  over  that  sphere  approaches  zero 
as  its  limit.     For 


Hence  when  ^  becomes  infinite  the  surface  integral  over  the 
large  sphere  approaches   zero  as  its  limit. 
For  the  small  sphere 

1  r 

V,  —  •  d  a  =  --  r  r2  sin  6  d  0  dA. 
rla  r3 

Hence  the  integral  over  that  sphere  becomes 

Fsin0  de  d<f>. 


THE  INTEGRAL   CALCULUS   OF  VECTORS          233 

Let  V  be  supposed  to  be  finite  and  continuous  at  the  point 
(#!,  yv  2X)  which  has  been  selected  as  origin.  Then  for  the 
surface  integral  V  is  practically  constant  and  equal  to  its 
value 

V  (ajj,  ylt  «!> 

at  the  point  in  question. 

sin  6  de  d  <    =  4  TT. 


f  f  s 


-  f  /Vsi 


Hence  -  sin  Q  d  0  d     = 


when  the  radius  R'  of  the  sphere  of  integration  approaches 
zero  as  its  limit.     Hence 


and  V-  VPotF=-47rF.  (52) 

In  like  manner  if  W  is  a  vector  function  which  has  in 
general  a  definite  potential,  then  that  potential  satisfies  Pois- 
son's  Equation. 

V  •  V  Pot  W  =  -  4  TT  W.  (52)' 

The  proof  of  this  consists  in  resolving  W  into  its  three  com- 
ponents.    For  each  component  the  equation  holds.     Let 


V  •  VPot^r=-47r  X, 
V«  VPotF=-47r  F, 
V  •  V  Pot  Z  —  -  4  TT  Z. 
Consequently 

V-  VPot(JTi+  Fj  +  ^k)  =  -47r(Xi+  Fj 


234  VECTOR  ANALYSIS 

Theorem :  IfVandW  are  such  functions  of  position  in  space 
that  their  potentials  exist  in  general,  then  for  all  points  at  which 
V  and  W  are  finite  and  continuous  those  potentials  satisfy 
Poisson's  Equation, 

V.  VPot  F=-47rF,  (52) 

V  •  V  Pot  W  =  -  4  TT  W.  (52)' 

The  modifications  in  this  theorem  which  are  to  be  made  at 
points  at  which  V  and  W  become  discontinuous  will  not  be 
taken  up  here. 
93.]     It  was  seen  (46)  Art.  91  that 

V-  VPotF=V.NewF=Max  VF. 
Hence  V  •  New  V  =  -  4  TT  V  (53) 

or  MaxVF=  —  4?rF. 

In  a  similar  manner  it  was  seen  (51)  Art.  91  that 

V  •  V  Pot  W  =  V  Max  W  -  V  x  Lap  W 
=  New  V  •  W  —  Lap  V  x  W. 

Hence         V  Max  W  —  V  x  Lap  W  =  —  4  TT  W,          (54) 
or  New  V  -  W  —  Lap  V  x  W  =  —  4  ?r  W.         (54)' 

By  virtue  of  this  equality  W  is  divided  into  two  parts. 

W  =  -r—  LapVxW  —  7—  NewV.W.         (55) 

4-7T  4?T 

Let  W  =  Wj  +  W2, 

1  1 

where       W ,  =  -r—  Lap  V  x  W  =  -. —  Lap  curl  W       (56) 
4?r  4?r 

and        W2  =  -  —  New  V  •  W  =  —  —  New  div  W.    (57) 


THE  INTEGRAL   CALCULUS   OF  VECTORS          235 

Equation  (55)  states  that  any  vector  function  W  multiplied 
by  4  TT  is  equal  to  the  difference  of  the  Laplacian  of  its  curl 
and  the  Newtonian  of  its  divergence.  Furthermore 

V  •  W,  =  —  V.LapV  x  W  =  -r-  V-VxLapW,. 
4?r  4?r 

But  the  divergence  of  the  curl  of  a  vector  function  is  zero. 
Hence  V.W1  =  divW1  =  0  (58) 

V  x  Wo  =  —  -r—  V  x  New  V  •  W2  =  —  -r—  VxV  Max  W2. 
4?r  4-n- 

But  the  curl  of  the  derivative  of  a  scalar  function  is  zero. 
Hence  V  x  W2  =  curl  W2  =  0.  (59) 

Consequently  any  vector  function  W  which  has  a  potential 
may  be  divided  into  two  parts  of  which  one  has  no  divergence 
and  of  which  the  other  has  no  curl.  This  division  of  W  into 
two  such  parts  is  unique. 

In  case  a  vector  function  has  no  potential  but  both  its  curl 
and  divergence  possess  potentials,  the  vector  function  may  be 
divided  into  three  parts  of  which  the  first  has  no  divergence ; 
the  second,  no  curl;  the  third,  neither  divergence  nor  curl. 

Let      W  =  — LapVxW-^-NewV«W  + W3.     (55)' 
As  before 

—  V  •  Lap  V  x  W  =  r—  V  •  V  x  Pot  V  x  W  =  0 
4?r  4-7T 

—  1  —1 

and      —  V  x  New  V  •  W  =  ^—  V  x  V  Pot  V  •  W  =  0. 
4-7T  4?r 

The  divergence  of  the  first  part  and  the  curl  of  the  second 
part  of  W  are  therefore  zero. 


236  VECTOR  ANALYSIS 

JLvxLapVxW  =  —  VxVxPotVxW 

47T  47T 

-  VV  •  Pot  V  x  W  —  T-  V  •  V  Pot  V  x  W. 
4?r  4?r 

—  VV  •  Pot  V  X  W  =  -:—  V  Pot  V  •  V  X  W  =  0, 

4-7T  4-7T 

for  V  •  V  x  W  =  0. 

Hence  —  V  •  VPot  V  x  W  =  V  X  W. 

4?r 

Hence      —  V  x  Lap  VxW  =  VxW  =  VxWr 

47T 

The  curl  of  W  is  equal  to  the  curl  of  the  first  part 
- —  Lap  V  x  W 

4  7T 

into  which  W  is  divided.  Hence  as  the  second  part  has  no 
curl,  the  third  part  can  have  none.  Moreover 

—  —  V  New  VW  =  V»W  =  V«W1. 
4?r 

Thus  the  divergence  of  W  is  equal  to  the  divergence  of 
the  second  part 

- —  New  V  •  W. 

4"7T 

into  which  W  is  divided.  Hence  as  the  first  part  has  no 
divergence  the  third  can  have  none.  Consequently  the  third 
part  W8  has  neither  curl  nor  divergence.  This  proves  the 
statement. 

By  means  of  Art.  96  it  may  be  seen  that  any  function  W3 
which  possesses  neither  curl  nor   divergence,  must  either 


THE  INTEGRAL   CALCULUS  OF   VECTORS          237 

vanish  throughout  all  space  or  must  not  become  zero  at 
infinity.  In  physics  functions  generally  vanish  at  infinity. 
Hence  functions  which  represent  actual  phenomena  may  be 
divided  into  two  parts,  of  which  one  has  no  divergence  and 
the  other  no  curl. 

94.]  Definition :  A  vector  function  the  divergence  of  which 
vanishes  at  every  point  of  space  is  said  to  be  solenoidal.  A 
vector  function  the  curl  of  which  vanishes  at  every  point  of 
space  is  said  to  be  irrotational. 

In  general  a  vector  function  is  neither  solenoidal  nor  irrota- 
tional. But  it  has  been  shown  that  any  vector  function  which 
possesses  a  potential  may  be  divided  in  one  and  only  one 
way  into  two  parts  Wls  W2  of  which  one  is  solenoidal  and 
the  other  irrotational.  The  following  theorems  may  be  stated. 
They  have  all  been  proved  in  the  foregoing  sections. 

With  respect  to  a  solenoidal  function  Wj,  the  operators 

- —  Lap  and  V  X  or  curl 
4?r 

are  inverse  operators.     That  is 

—  Lap  V  x  Wj  =  V  x  ?—  Lap  Wx  =  Wr  (60) 

4?r  4?r 

Applied  to  an  irrotational  function  W2  either  of  these  opera- 
tors gives  zero.  That  is 

—  Lap  W2  =  0  ,  V  x  W2  =  0. 
With  respect  to  an  irrotational  function  W2,  the  operators 

- —  New  and  —  v  or  — •  div 

4-7T 

are  inverse  operators.     That  is 

_  JL  New  V  .  W2  =  -  V  •  ~  New  W2  =  W2.    (62) 

4  7T  4:  7T 


238  VECTOR  ANALYSIS 

With  respect  to  a  scalar  function  V  the  operators 

—  V  •  or  —  div  and  -  -  New, 

4?T 

and  also  —  ~r—  Max  and  V 

4-7T 

are  inverse  operators.    That  is 


=F  (63) 

4-7T 

and  -J-MaxVT=F. 

4?r 

JFtYA  respect  to  a  solenoidal  function  Wj  tf^e  operators 

-  —  Pot  awo*  V  X  V  X  or  curl  curl 

4?r 

are  inverse  operators.     That  is 

—  Pot  V  x  V  x  Wj  =  V  x  V  x  —  Pot  Wi  =  Wr  (64) 

4-7T  4  7T 

fF^A  respect  to  an  irrotational  function  W2  the  operators 
—  Pot  aw(i  —  VV  • 

4"7T 

are  inverse  operators.     That  is 

-    -  Pot  VV  .  W2  =  -  VV  .        Pot  W2  =  W2.  (65) 


With  respect  to  any  scalar  or  vector  function  V,  W  the 
operators 

•r—  Pot  and  —  V  •  V 
4?r 

are  mwrse  operators.     That  is 


THE  INTEGRAL   CALCULUS  OF   VECTORS          239 

_  -—Pot  V- VF=-  V.  V  —  PotF=F 
4?r  4?r 

and     -  —  Pot  V  .  V  W  =  -  V  •  V  —  Pot  W  =  W.  (66) 

4-7T  47T 

With  respect  to  a  solenoidal  function  Wj  the  differentiating 
operators  of  the  second  order 

—  V  •  V  and  V  X  V  x 
are  equivalent 

-  V  •  V  Wj  =  V  x  V  x  Wr  (67) 

With  respect  to  an  irrotational  function  W2  the  differentiat- 
ing operators  of  the  second  order 

V  •  V  and  V  V  • 
are  equivalent.     That  is 

V»VW2  =  VV»  W2.  (68) 

By  integrating  the  equations 

47rF=- V.NewF 
and  4  TT  W  =  V  x  Lap  W  —  V  Max  W 

by  means  of  the  potential  integral  Pot 

4  TT  Pot  F=  -  Pot  V  •  New  F=  -  Max  New  F    (69) 
47T  Pot  W  =  Pot  V  x  Lap  W  -  Pot  V  Max  W 

4  TT  Pot  W  =  Lap  Lap  W  —  New  Max  W.     (70) 
Hence  for  scalar  functions  and  irrotational  vector  functions 

—  - —  New  Max 
4?r 

is  an  operator  which  is  equivalent  to  Pot.    For  solenoidal  vector 
functions  the  operator          .. 

-  Lap  Lap 

4  7T 


240  VECTOR  ANALYSIS 

gives  the  potential.  For  any  vector  function  the  first  operator 
gives  the  potential  of  the  irrotational  part;  the  second,  the 
potential  of  the  solenoidal  part. 

*95.]  There  are  a  number  of  double  volume  integrals  which 
are  of  such  frequent  occurrence  in  mathematical  physics  as 
to  merit  a  passing  mention,  although  the  theory  of  them  will 
not  be  developed  to  any  considerable  extent.  These  double 
integrals  are  all  scalar  quantities.  They  are  not  scalar  func- 
tions of  position  in  space.  They  have  but  a  single  value. 
The  integrations  in  the  expressions  may  be  considered  for 
convenience  as  extended  over  all  space.  The  functions  by 
vanishing  identically  outside  of  certain  finite  limits  deter- 
mine for  all  practical  purposes  the  limits  of  integration  in 
case  they  are  finite. 

Given  two  scalar  functions  ?7,  V  of  position  in  space. 
The  mutual  potential  or  potential  product,  as  it  may  be  called, 
of  the  two  functions  is  the  sextuple  integral 


Pot 

(71) 

One  of  the  integrations  may  be  performed 

Pot  (  U,  F)  =fff  PX*!,  yi,  *i)  Pot  Vd  Vl 

(*2,  yv  *2)  Pot  Udv2.  (72) 


In  a  similar  manner  the  mutual  potential  or  potential  product 
of  two  vector  functions  W,  W"  is 


(71)' 
This  is  also  a  scalar  quantity.     One  integration  may  be  car- 


ried out 


THE  INTEGRAL   CALCULUS   OF  VECTORS          241 
Pot  (W,  W")  =  W'0*i'  2/i'  *i)  '  Pot  W"  dvi 


The    mutual    Laplacian    or    Laplacian    product    of   two 
vector  functions  W',  W"  of  position  in  space  is  the  sextuple 

integral 

Lap(W',W") 

=ffffffw'  (*i»  ?!•  2i)-^  x  w"  (*«»  y*»  22)  rf«i  ^^2- 

(73) 

One  integration  may  be  performed. 

Lap  (W,  W")  =f  f  fw"  (xv  yv  «,)  •  Lap  W  d  vz 

(74) 

d  v- 


=fffw'  fa*  y»  ^i)  •  LaP  w" 


The  Newtonian  product  of  a  scalar  function  F",  and  a  vector 
function  W  of  position  in  space  is  the  sextuple  integral 


(75) 

By  performing  one  integration 


vz.    (76) 


In  like  manner  the  Maxwellian  product  of  a  vector  function 
W  and  a  scalar  function  F  of  position  in  space  is  the 
integral 

Max  (W,F)  = 

(77) 

16 


242  VECTOR  ANALYSIS 

One  integration  yields 

Max(W,  r)= 


(78) 
By  (53)  Art.  93. 

4  TT  U  PotF=  -  (V  •  New  *7)  PotF. 

V  •  [NewZ7PotF]  =(V  •  New  U)  Pot  F  +  (New  Z7)  .  V  Pot  F. 
-  (V  •  New  U)  PotF=  -  V  .  [New  Z7PotF]+  New  U>  New  V. 
Integrate  : 
47r  f  f  CuPotYdv=:-  f  f  fv-  [Ne 

+  f  f  f  NewZ7  •  NewF  ci  v. 
4-7T  Pot  (Z7,  F)  =  T  r  fNewZ 

-  f  f  Pot  F  New  CT  .  d  a.          (79) 

The  surface  integral  is  to  be  taken  over  the  entire  surface  S 
bounding  the  region  of  integration  of  the  volume  integral. 
As  this  region  of  integration  is  "  all  space,"  the  surface  S  may 
be  looked  upon  as  the  surface  of  a  large  sphere  of  radius  E. 
If  the  functions  U  and  V  vanish  identically  for  all  points  out- 
side of  certain  finite  limits,  the  surface  integral  must  vanish. 
Hence 

4  TT  Pot  (  U,  F)  =  f  f  fNew  U  •  New  Vdv.     (79)' 

By  (54)  Art.  93, 

47r  W".  PotW  =  V  x  Lap  W"  •  Pot  W 
-  V  Max  W"  •  Pot  W'. 


THE  INTEGRAL  CALCULUS  OF   VECTORS          243 

But    V  •  [Lap  W"  x  Pot  W]  =  Pot  W  -  V  x  Lap  W" 

-  Lap  W"  •  V  x  Pot  W', 

and     V  •  [Max  W"  Pot  W]  =  Pot  W  •  V  Max  W" 

+  Max  W"  V  •  Pot  W. 

Hence  V  x  Lap  W"  •  Pot  W  =  V  •  [Lap  W"  x  Pot  W] 

+  Lap  W"  •  Lap  W, 

and       V  Max  W"  •  Pot  W'  =  V  •  [Max  W"  Pot  W] 

—  Max  W"  Max  W. 
Hence  substituting: 

4  TT  W"  •  Pot  W  =  Lap  W  •  Lap  W  +  Max  W  Max  W" 
+  V  •  [Lap  W"  X  Pot  W] 

-V-  [Max  W"  Pot  W]. 

Integrating: 

4  TT  Pot  (W,  W")  =  f  f  f  Lap  W'.  Lap  W'dv 

+  f  f  fMaxW'MaxW"rfv   (80) 

-  f  f  PotW'  x  Lap  W"  da-  f  f  Max  W"  Pot  WW  a. 

If  now  W'  and  W"  exist  only  in  finite  space  these  surface 
integrals  taken  over  a  large  sphere  of  radius  R  must  vanish 
and  then 

4  TT  Pot  (W',  W")  =ff  /Lap  W'  •  Lap  W"  d  v 

+  f  f  |*Max  W'  Max  W"  d  v.         (80)' 

*  96.]    There  are  a  number  of  useful  theorems  of  a  function- 
theoretic  nature  which  may  perhaps  be  mentioned  here  owing 


244  VECTOR  ANALYSIS 

to  their  intimate  connection  with  the  integral  calculus  of 
vectors.  The  proofs  of  them  will  in  some  instances  be  given 
and  in  some  not.  The  theorems  are  often  useful  in  practical 
applications  of  vector  analysis  to  physics  as  well  as  in  purely 
mathematical  work. 

Theorem  :  If  V  (x,  y,  z)  be  a  scalar  function  of  position 
in  space  which  possesses  in  general  a  definite  derivative  V  V 
and  if  in  any  portion  of  space,  finite  or  infinite  but  necessarily 
continuous,  that  derivative  vanishes,  then  the  function  V  is 
constant  throughout  that  portion  of  space. 

Given  VF=0. 

To  show  F=  const. 

Choose  a  fixed  point  (xv  yv  zj  in  the  region.  By  (2)  page 
180 

fxt  *  *    V  V*  d  r  =  Ffa  y,  *)  -  F  fa,  yv  *,)• 

J  *u  Vi,  *i 


But 

Hence  F(#,  y,  z)  =  F  (xv  yv  Zj)  =  const. 

Theorem  :  If  F  (#,  y,  2)  be  a  scalar  function  of  position 
in  space  which  possesses  in  general  a  definite  derivative  V  V  '  ; 
if  the  divergence  of  that  derivative  exists  and  is  zero  through- 
out any  region  of  space,1  finite  or  infinite  but  necessarily 
continuous  ;  and  if  furthermore  the  derivative  V  F  vanishes 
at  every  point  of  any  finite  volume  or  of  any  finite  portion  of 
surface  in  that  region  or  bounding  it,  then  the  derivative 
vanishes  throughout  all  that  region  and  the  function  F  re- 
duces to  a  constant  by  the  preceding  theorem. 

1  The  term  throughout  any  region  of  space  must  be  regarded  as  including  the 
boundaries  of  the  region  as  well  as  the  region  itself. 


THE  INTEGRAL   CALCULUS  OF  VECTORS          245 

Given  V  •  VF"=  0  for  a  region  T, 

and  VF=  0  for  a  finite  portion  of  surface  S. 

To  show  F=  const. 

Since  VF  vanishes  for  the  portion  of  surface  S,  Fis  certainly 
constant  in  S.  Suppose  that,  upon  one  side  of  S  and  in  the 
region  T,  V  were  not  constant.  The  derivative  V  V  upon 
this  side  of  S  has  in  the  main  the  direction  of  the  normal  to 
the  surface  S.  Consider  a  sphere  which  lies  for  the  most 
part  upon  the  outer  side  of  S  but  which  projects  a  little 
through  the  surface  S.  The  surface  integral  of  VF"  over 
the  small  portion  of  the  sphere  which  projects  through  the 
surface  S  cannot  be  zero.  For,  as  V  V  is  in  the  main  normal 
to  $,  it  must  be  nearly  parallel  to  the  normal  to  the  portion 
of  spherical  surface  under  consideration.  Hence  the  terms 

VF.  rfa 

in  the  surface  integral  all  have  the  same  sign  and  cannot 
cancel  each  other  out.  The  surface  integral  of  VF  over 
that  portion  of  S  which  is  intercepted  by  the  spherical  sur- 
face vanishes  because  V  F  is  zero.  Consequently  the  surface 
integral  of  V  F  taken  over  the  entire  surface  of  the  spherical 
segment  which  projects  through  S  is  not  zero. 

But          f  f  VF.<£a  =  f  f  fv»  VFd0  =  0. 

Hence  f  fvF'da  =  0. 

It  therefore  appears  that  the  supposition  that  F  is  not 
constant  upon  one  side  of  S  leads  to  results  which  contradict 
the  given  relation  V  •  V  F  =  0.  The  supposition  must  there- 
fore have  been  incorrect  and  F  must  be  constant  not  only  in 
S  but  in  all  portions  of  space  near  to  $  in  the  region  T.  By 


246  VECTOR  ANALYSIS 

an  extension  of  the  reasoning  V  is  seen  to  be  constant 
throughout  the  entire  region  T. 

Theorem :  If  V  (x,  y,  z)  be  a  scalar  function  of  position  in 
space  possessing  in  general  a  derivative  V  V  and  if  through- 
out a  certain  region 1  T  of  space,  finite  or  infinite,  continuous 
or  discontinuous,  the  divergence  V  •  V  V  of  that  derivative 
exists  and  is  zero,  and  if  furthermore  the  function  V  possesses 
a  constant  value  c  in  all  the  surfaces  bounding  the  region 
and  V(x,  y,  z)  approaches  c  as  a  limit  when  the  point  (x,  y,  z) 
recedes  to  infinity,  then  throughout  the  entire  region  T  the 
function  V  has  the  same  constant  value  c  and  the  derivative 
VF  vanishes. 

The  proof  does  not  differ  essentially  from  the  one  given 
in  the  case  of  the  last  theorem.  The  theorem  may  be  gen- 
eralized as  follows : 

Tlieorem :  If  V(x,  y,  z)  be  any  scalar  function  of  position 
in  space  possessing  in  general  a  derivative  V  V;  if  U  (#,  y,  z) 
be  any  other  scalar  function  of  position  which  is  either  posi- 
tive or  negative  throughout  and  upon  the  boundaries  of  a 
region  T,  finite  or  infinite,  continuous  or  discontinuous;  if 
the  divergence  V  •  [  U  V  F]  of  the  product  of  U  and  V  F 
exists  and  is  zero  throughout  and  upon  the  boundaries  of  T 
and  at  infinity ;  and  if  furthermore  V  be  constant  and  equal 
to  c  upon  all  the  boundaries  of  T  and  at  infinity ;  then  the 
function  V  is  constant  throughout  the  entire  region  T  and 
is  equal  to  c. 

Theorem :  If  F  (x,  y,  z)  be  any  scalar  function  of  position 
in  space  possessing  in  general  a  derivative  V  F;  if  through- 
out any  region  T  of  space,  finite  or  infinite,  continuous  or 
discontinuous,  the  divergence  V  •  V  F  of  this  derivative  exists 
and  is  zero ;  and  if  in  all  the  bounding  surfaces  of  the  region 
T  the  normal  component  of  the  derivative  V  F  vanishes  and 
at  infinite  distances  in  T  (if  such  there  be)  the  product 

1  The  region  includes  its  boundaries. 


THE  INTEGRAL   CALCULUS  OF  VECTORS         247 

rz  9  V/  9  r  vanishes,  where  r  denotes  the  distance  measured 
from  any  fixed  origin ;  then  throughout  the  entire  region  T 
the  derivative  VF  vanishes  and  in  each  continuous  portion 
of  T  V  is  constant,  although  for  different  continuous  portions 
this  constant  may  not  be  the  same. 

This  theorem  may  be  generalized  as  the  preceding  one 
was  by  the  substitution  of  the  relation  V  •  (  U  V  F)  =  0  for 
V-VF=Oand  Urz9V/9r  =  0  for  r29V/9r  =  0. 

As  corollaries  of  the  foregoing  theorems  the  following 
statements  may  be  made.  The  language  is  not  so  precise 
as  in  the  theorems  themselves,  but  will  perhaps  be  under- 
stood when  they  are  borne  in  mind. 

If  V  U  =  V  F,  then  U  and  V  differ  at  most  by  a 
constant. 

If  V«V?7=V.VF  and  if  VZ7  =  VFin  any  finite 
portion  of  surface  $,  then  V  U  =  V  V  at  all  points  and  V 
differs  from  V  only  by  a  constant  at  most. 

If  V»VZ7=  V.VF  and  if  U=  V  in  all  the  bounding 
surfaces  of  the  region  and  at  infinity  (if  the  region  extend 
thereto),  then  at  all  points  £7" and  Fare  equal. 

If  V  •  V  £7  =  V  •  V  F  and  if  in  all  the  bounding  surfaces 
of  the  region  the  normal  components  of  V  U  and  V  F  are 
equal  and  if  at  infinite  distances  rz(9  U /9 r  —  9  V/9 r)  is 
zero,  then  V  27 and  V  Fare  equal  at  all  points  of  the  region 
and  U  differs  from  F  only  by  a  constant. 

Theorem :  If  W '  and  W  "  are  two  vector  functions  of  position 
in  space  which  in  general  possess  curls  and  divergences ;  if 
for  any  region  T,  finite  or  infinite  but  necessarily  continuous, 
the  curl  of  W'  is  equal  to  the  curl  of  W"  and  the  divergence 
of  W'  is  equal  to  the  divergence  of  W";  and  if  moreover 
the  two  functions  W'  and  W"  are  equal  to  each  other  at 
every  point  of  any  finite  volume  in  T  or  of  any  finite  surface 
in  T  or  bounding  it;  then  W'  is  equal  to  W"  at  every  point 
of  the  region  T. 


248  VECTOR  ANALYSIS 

Since  V  x  W  =  V  x  W",  V  x  (W  -  W")  =  0.  A  vec- 
tor function  whose  curl  vanishes  is  equal  to  the  derivative 1 
of  a  scalar  function  V  (page  197).  Let  VF=W'-  W". 
Then  V  •  V  V—  0  owing  to  the  equality  of  the  divergences. 
The  theorem  therefore  becomes  a  corollary  of  a  preceding  one. 

Theorem  :  If  W'  and  W"  are  two  vector  functions  of  posi- 
tion which  in  general  possess  definite  curls  and  divergences  ; 
if  throughout  any  aperiphractic2  region  T,  finite  but  not 
necessarily  continuous,  the  curl  of  W'  is  equal  to  the  curl  of 
W"  and  the  divergence  of  W'  is  equal  to  the  divergence  of 
W";  and  if  furthermore  in  all  the  bounding  surfaces  of  the 
region  T  the  tangential  components  W'  and  W"  are  equal; 
then  W'  is  equal  to  W"  throughout  the  aperiphractic  region  T. 

Theorem,:  If  W'  and  W"  are  two  vector  functions  of  posi- 
tion in  space  which  in  general  possess  definite  curls  and 
divergences  ;  if  throughout  any  acyclic  region  T,  finite  but  not 
necessarily  continuous,  the  curl  of  W'  is  equal  to  the  curl 
W"  and  the  divergence  of  W'  is  equal  to  the  divergence  of 
W";  and  if  in  all  the  bounding  surfaces  of  the  region  T  the 
normal  components  of  W'  and  W"  are  equal ;  then  the  func- 
tions W'  and  W"  are  equal  throughout  the  region  acyclic  T. 

The  proofs  of  these  two  theorems  are  carried  out  by  means 
of  the  device  suggested  before. 

Theorem:  If  W'  and  W"  are  two  vector  functions  such 
that  V  -  V  W'  and  V  •  V  W"  have  in  general  definite  values 
in  a  certain  region  T,  finite  or  infinite,  continuous  or  discon- 
tinuous ;  and  if  in  all  the  bounding  surfaces  of  the  region 
and  at  infinity  the  functions  W'  and  W"  are  equal ;  then  W' 
is  equal  to  W"  throughout  the  entire  region  T. 

The  proof  is  given  by  treating  separately  the  three  com- 
ponents of  W'  and  W". 

1  The  region  T  may  hare  to  be  made  acyclic  by  the  insertion  of  diaphragms. 
s  A  region  which  encloses  within  itself  another  region  is  said  to  be  periphrac- 
tic.    If  it  encloses  no  region  it  is  aperiphractic. 


THE  INTEGRAL    CALCULUS  OF  VECTORS         249 


SUMMARY  OF  CHAPTER  IV 

The  line  integral  of  a  vector  function  W  along  a  curve  C  is 
defined  as 

C  w<Zr=f  [Widx  +  Wtdy  +  Wtdz].       (1) 

J  C  «/  C 


The  line  integral  of  the  derivative  V  V  of  a  scalar  function 
V  along  a  curve  C  from  r0  to  r  is  equal  to  the  difference 
between  the  values  of  V  at  the  points  r  and  r0  and  hence  the 
line  integral  taken  around  a  closed  curve  is  zero  ;  and  con- 
versely if  the  line  integral  of  a  vector  function  W  taken 
around  any  closed  curve  vanishes,  then  W  is  the  derivative 
V  V  of  some  scalar  function  V. 


=  F(r)-F(r0)  (2) 

.rfr  =  0  (3) 


fvF.rfr  =  0 
J  o 

and  if  f  W  •  dr  =  0,  then  W  =  VF. 

Jo 

Illustration  of  the  theorem  by  application  to  mechanics. 

The  surface  integral  of  a  vector  function  W  over  a  surface 
S  is  denned  as 

f  f  W  •  d&=  C  C  [Wldy  dz  +  W^dz  dx  +  Wzdx  dy]. 

Gauss's  Theorem  :  The  surface  integral  of  a  vector  func- 
tion taken  over  a  closed  surface  is  equal  to  the  volume 
integral  of  the  divergence  of  that  function  taken  throughout 
the  volume  enclosed  by  that  surface 

f  f  JV  •  W  dv  =  C  C  W  •  da,  (7) 


250  VECTOR  ANALYSIS 


1     '     '  l   *  ~    '    3y 


s 


[X  dy  dz  +  Y  dz  dx  +  Z  dx  dy~],  (8) 


if  X,  I7,  ^  be  the  three  components  of  the  vector  function  W. 
Stokes's  Theorem:  The  surface  integral  of  the  curl  of  a 
vector  function  taken  over  any  surface  is  equal  to  the  line 
integral  of  the  function  taken  around  the  line  bounding  the 
surface.  And  conversely  if  the  surface  integral  of  a  vector 
function  TJ  taken  over  any  surface  is  equal  to  the  line  integral 
of  a  function  W  taken  around  the  boundary,  then  U  is  the 
curl  of  W. 

f  f  V  x  W-da=  f  W»rfr,  (11) 

J  J  a  Jo 

and  if      J  J  JJ  •  d  a  =J  W  •  dr,  then  U  =  V  X  W.         (12) 

Application  of  the  theorem  of  Stokes  to  deducing  the 
equations  of  the  electro-magnetic  field  from  two  experimental 
facts  due  to  Faraday.  Application  of  the  theorems  of  Stokes 
and  Gauss  to  the  proof  that  the  divergence  of  the  curl  of 
a  vector  function  is  zero  and  the  curl  of  the  derivative  of 
a  scalar  function  is  zero. 

Formulae  analogous  to  integration  by  parts 


/  u  V  v  •  d  T  =  [u  v"fr     -Tv^U'dr,  (14) 

x  v'd&  =fo  uv'  ff*-Jj^»VXY»rfl,(M) 

dr,    (16) 


THE  INTEGRAL   CALCULUS  OF   VECTORS          251 
I  I    f  w  V  •  v  dv  =  /    /    w  v  •  d&  —   I   I   I^U'vdv,    (17) 

f  f  VM  x  vrfa  =  -  f  f  TV**-  V  x  vdv.     (18) 

Green's  Theorem: 
I   I    IV  u  •  V  v  dv  =  If  uVvd&—  i  I    iu  V  •  V  v  dv 

=  CCv  Vw  •  da  —  I    f  Cv  V  •  Vudv,          (19) 
C  C  C(uV'Vv  —  vV^Vw)  dv=  I   MwVv  —  v  Vw)^a.  (20) 

Kelvin's  generalization: 
|  I  I  wVu*Vvdv=  I    I  uw*7v»d&—  I  I  I  u*  V[wVv]  c?v 

=  I    fvw  Vu*da,—  i  I    Cv  V»  [w  VM]  dv.      (21) 

The  integrating  operator  known  as  the  potential  is  defined 
by  the  equation 

=f  f  f  V(X*  Vv  Zz)  dxz  dy,  dzv  (22) 

J  J  J  r 


Pot  V 


Pot  W  -a;2y2^2    <**2  ^2/2  ^^2-        (23) 


VPotF=PotVF;  (27) 

V  x  Pot  W  =  Pot  V  x  W,  (28) 

V  .  Pot  W  =  Pot  V  .  W,  (29) 

V  •  V  Pot  V=  Pot  V  •  Vr,  (30) 


252  VECTOR  ANALYSIS 

V  •  V  Pot  W  =  Pot  V  •  V  W,  (31) 

VV  •  Pot  W  =  Pot  W  •  W,  (32) 

V  x  V  x  Pot  W  =  Pot  V  x  V  x  W.  (33) 

The  integrating  operator  Pot  and  the  differentiating  operator 
V  are  commutative. 

The  three  additional  integrating  operators  known  as  the 
Newtonian,  the  Laplacian,  and  the  Maxwellian. 

New  F=ri2F*2)  <**2  dy,  dz2.  (42) 


T          TIT  VT  i/2'     2x    j         j         j  /-Af>\ 

Lap  W  =  I   I   j  — -a —         2  ^^2  *ir      (43) 

^12 


MaxW  = 


If  the  potential  exists  these  integrals  are  related  to  it  as  fol- 
lows: 

V  Pot  F=  New  F, 

V  x  Pot  W  =  Lap  W;  (45) 

V  •  Pot  W  =  Max  W. 

The  interpretation  of  the  physical  meaning  of  the  Newtonian 
on  the  assumption  that  F  is  the  density  of  an  attracting 
body,  of  the  Laplacian  on  the  assumption  that  W  is  electric 
flux,  of  the  Maxwellian  on  the  assumption  that  W  is  the 
intensity  of  magnetization.  The  expression  of  these  integrals 
or  their  components  in  terms  of  #,  y,  z  ;  formulae  (42)',  (43)', 
(44)'  and  (42)",  (43)",  (44)". 

V  •  New  F=  Max  V  F,  (46) 

V  Max  W  =  New  V  •  W,  (47) 

V  x  Lap  W  =  Lap  V  x  W,  (48) 


THE  INTEGRAL   CALCULUS  OF  VECTORS          253 
V  •  Lap  W  =  Max  V  x  W  =  0,  (49) 

V  x  New  V  =  Lap  V  V-  0,  (50) 

V  •  V  W  =  New  V  •  W  —  Lap  V  x  W 

=  V  Max  W  —  V  x  Lap  W.          (51) 

The  potential  is  a  solution  of  Poisson's  Equation.    That  is, 

V-  VPotF  =  -47rF,  (52) 

and  V  •  V  Pot  F=  -  4  TT  W.  (52)' 

F=^V.NewF,  (53) 

4  IT 

W  =  -. —  Lap  V  x  W  —  -. —  New  V  •  W.       (55) 

4-7T  4  7T 

Hence  W  is  divided  into  two  parts  of  which  one  is 
solenoidal  and  the  other  irrotational,  provided  the  potential 
exists.  In  case  the  potential  does  not  exist  a  third  term  W3 
must  be  added  of  which  both  the  divergence  and  the  curl 
vanish.  A  list  of  theorems  which  follow  immediately  from 
equations  (52),  (52)',  (53),  (55)  and  which  state  that  certain 
integrating  operators  are  inverse  to  certain  differentiating 
operators.  Let  V  be  a  scalar  function,  W,  a  solenoidal  vector 
function,  and  W2  an  irrotational  vector  function.  Then 

j— Lap  V  x  Wt  =  Vx  j— LapWj  =  Wr    (60) 

—  Lap  W2  =  0,     V  x  W2  =  0  (61) 

-  —  New  V  •  W2  =  -  V  •  —  New  W2  =  W2.  (62) 

47T  4  7T 


254  VECTOR  ANALYSIS 

V.  -^NewF=  V 


(63) 


-        Max  VF  =  F. 

47T 


--  Pot  V  x  V  x  Wi  =  V  x  V  x  Pot  Wj  =  Wx      (64) 

47T 

_-Lp0t  VV.W2  =  -  V.V--PotW2  =  W2.  (65) 

4  7T  4-7T 


_i  Pot  V 

4-7T  4-Tr 

1  1  (66) 

_  -—Pot  V  •  V  W  =  -  V  •  V  —  Pot  W  =  W. 
4?r  4?r 

-V.VW1  =  VxVxW1  (67) 

V  •  V  W2  =  VV  •  W2  (68) 

4  TT  Pot  V  =  -  Max  New  V  (69) 

4  TT  Pot  W  =  Lap  Lap  W  —  New  Max  W.  (70) 

Mutual  potentials  Newtonians,  Laplacians,  and  Maxwellians 
may  be  formed.  They  are  sextuple  integrals.  The  integra- 
tions cannot  all  be  performed  immediately  ;  but  the  first  three 
may  be.  Formulae  (71)  to  (80)  inclusive  deal  with  these  inte- 
grals. The  chapter  closes  with  the  enunciation  of  a  number 
of  theorems  of  a  function-theoretic  nature.  By  means  of 
these  theorems  certain  facts  concerning  functions  may  be 
inferred  from  the  conditions  that  satisfy  Laplace's  equation 
and  have  certain  boundary  conditions. 

Among  the  exercises  number  6  is  worthy  of  especial  atten- 
tion. The  work  done  in  the  text  has  for  the  most  part  assumed 
that  the  potential  exists.  But  many  of  the  formulae  connecting 
Newtonians,  Laplacians,  and  Maxwellians  hold  when  the  poten- 
tial does  not  exist.  These  are  taken  up  in  Exercise  6  referred  to. 


THE  INTEGRAL   CALCULUS  OF  VECTORS         255 

EXEKCISES  ON  CHAPTER  IV 

I.1  If  F"  is  a  scalar  function  of  position  in  space  the  line 
integral 

«/><" 

is  a  vector  quantity.     Show  that 

r  c  c    v  c  r 

Jo  «/     i/  5  J    tjs 

That  is  ;  the  line  integral  of  a  scalar  function  around  a 
closed  curve  is  equal  to  the  skew  surface  integral  of  the  deriv- 
ative of  the  function  taken  over  any  surface  spanned  into 
the  contour  of  the  curve.  Show  further  that  if  V  is  constant 
the  integral  around  any  closed  curve  is  zero  and  conversely 
if  the  integral  around  any  closed  curve  is  zero  the  function  V 
is  constant. 

Hint :  Instead  of  treating  the  integral  as  it  stands  multiply 
it  (with  a  dot)  by  an  arbitrary  constant  unit  vector  and  thus 
reduce  it  to  the  line  integral  of  a  vector  function. 

2.   If  W  is  a  vector  function  the  line  integral 


~Jc 


W  X  di 
c 


is  a  vector  quantity.  It  may  be  called  the  skew  line  integral 
of  the  function  W.  If  c  is  any  constant  vector,  show  that  if 
the  integral  be  taken  around  a  closed  curve 


1  The  first  four  exercises  are  taken  from  Foppl'a  Einfiihrung  in  die  Max- 
well'sche  Theorie  der  Electricitat  where  they  are  worked  out. 


256  VECTOR  ANALYSIS 

and     H  •  c  =  o  •   |J  js  V  •  W  d  a  -  J  j  g  V  (W  •  d  a)  j 


In  case   the  integral  is  taken  over  a  plane  curve   and  the 
surface  S  is  the  portion  of  plane  included  by  the  curve 


=  ffa[ 


-  V 


Show  that  the  integral  taken  over  a  plane  curve  vanishes 
when  W  is  constant  and  conversely  if  the  integral  over  any 
plane  curve  vanishes  W  must  be  constant. 

3.  The  surface  integral  of  a  scalar  function  V  is 

•  = 

This  is  a  vector  quantity.  Show  that  the  surface  integral 
of  V  taken  over  any  closed  surface  is  equal  to  the  volume 
integral  of  VF"  taken  throughout  the  volume  bounded  by 
that  surface.  That  is 


Hence  conclude  that  the  surface  integral  over  a  closed  sur- 
face vanishes  if  V  be  constant  and  conversely  if  the  surface 
integral  over  any  closed  surface  vanishes  the  function  V  must 
be  constant. 
4.   If  W  be  a  vector  function,  the  surface  integral 


=  /   I 


da.  x  W 

may  be   called  the  skew  surface   integral.     It  is  a   vector 
quantity.     Show  that  the  skew  surface  integral  of  a  vector 


THE  INTEGRAL   CALCULUS  OF  VECTORS         257 

function  taken  over  a  closed  surface  is  equal  to  the  volume 
integral  of  the  vector  function  taken  throughout  the  volume 
bounded  by  the  surface.  That  is 


C  C 


d&x  W  =  x  Wdv. 


Hence  conclude  that  the  skew  surface  integral  taken  over 
any  surface  in  space  vanishes  when  and  only  when  W  is  an 
irrotational  function.  That  is,  when  and  only  when  the  line 
integral  of  W  for  every  closed  circuit  vanishes. 

5.  Obtain   some   formulae   for  these   integrals  which  are 
analogous  to  integrating  by  parts. 

6.  The  work  in  the  text  assumes  for  the  most  part  that  the 
potentials  of  V  and  W  exist.    Many  of  the  relations,  however, 
may  be  demonstrated  without  that  assumption.     Assume  that 
the  Newtonian,  the  Laplacian,  the  Maxwellian  exist.      For 
simplicity  in  writing  let 


12 
-r 

2  r  12 


Then       New  V  =fff  ^i  Pu  ^(ajr  yv  *,)  d  vv        (81) 
Lap  W  =ViPu  X  W  (xv  yv  aa)  dvv     (82) 


Max  W  -Vtfu  «W  (xv  7/2,  «a)  dvv       (83) 

(84) 


17 


258  VECTOR  ANALYSIS 

By  exercise  < 

It  can  be  shown  that  if  F  is  such  a  function  that  New  V 
exists,  then  this  surface  integral  taken  over  a  large  sphere  of 
radius  R  and  a  small  sphere  of  radius  R'  approaches  zero 
when  R  becomes  indefinitely  great;  and  R',  indefinitely 
small.  Hence 


or  NewF=PotVF.  (85) 

Prove  in  a  similar  manner  that 

Lap  W  =  Pot  V  x  W,  (86) 

Max  W  =  Pot  V  •  W.  (87) 

By  means  of  (85),  (86),  (87)  it  is  possible  to  prove  that 
V  x  Lap  W  =  Lap  V  x  W, 
V«New  F=Max  VF, 

V  Max  W  =  New  V  •  W. 
Then  prove 

/*     /"»     />  /»     s+     s* 

VxLapW=/   /    l.plaVV.W  dv2  -  I    I    /j 
t/  t/  *J  »/  c/  t/ 

and  VMaxW=  f  C  fpn  VV»W  dv, 

Hence  V  x  Lap  W  —  V  Max  W  =  —  If   rPi»  V  •  V  W  rf  »* 

•*•  I    -*•     14  -* 

J  J  J 

Hence  V  x  Lap  W  —  V  Max  W  =  4  IT  W.  (88) 

7.   An  integral  used  by  Helmholtz  is 

o  Vdva. 


v 


THE  INTEGRAL   CALCULUS  OF  VECTORS          259 
or  if  W  be  a  vector  function 

zWdVr  (90) 


Show  that  the  integral  converges  if  F"  diminishes  so  rapidly 

that 

K 


when  r  becomes  indefinitely  great. 

V  H  (  F)  =  H  (V  F)  =  New  (r2  F),  (91) 

V  •  #  (W)  =  #  (V  •  W)  =  Max  (r2  W),  (92) 

V  x  #  (W)  =  H  (V  x  W)  =  Lap  (r2  W),  (93) 

V  •  VJ2"  (  F)  =  JT  (V  •  VF)=Max(r2VF)  =  2  Pot  ^  C94) 

V  •  V  #  (W)  =  H(y  •  V  W)  =  2  Pot  W.  (95) 


H  (  F)  =-        Pot  Pot  F:  (96) 


7T 


JET  (W)  =  -  ~  Pot  Pot  W.  (97) 


VV.J5"  (W).        (98) 

8.  Give  a  proof  of  Gauss's  Theorem  which  does  not  depend 
upon  the  physical  interpretation  of  a  function  as  the  flux  of  a 
fluid.     The  reasoning  is  similar  to  that  employed  in  Art.  51 
and  in  the  first  proof  of  Stokes's  Theorem. 

9.  Show  that  the  division  of  W  into  two  parts,  page  235, 
is  unique. 

10.  Treat,  in  a  manner  analogous  to  that  upon  page  220, 
the  case  in  which  V  has  curves  of  discontinuities. 


CHAPTER  V 

LINEAR  VECTOR  FUNCTIONS 

97.]  AFTER  the  definitions  of  products  had  been  laid  down 
and  applied,  two  paths  of  advance  were  open.  One  was 
differential  and  integral  calculus ;  the  other,  higher  algebra 
in  the  sense  of  the  theory  of  linear  homogeneous  substitutions. 
The  treatment  of  the  first  of  these  topics  led  to  new  ideas 
and  new  symbols  —  to  the  derivative,  divergence,  curl,  scalar 
and  vector  potential,  that  is,  to  V,  V«,  Vx,  and  Pot  with  the 
auxiliaries,  the  Newtonian,  the  Laplacian,  and  the  Maxwellian. 
The  treatment  of  the  second  topic  will  likewise  introduce 
novelty  both  in  concept  and  in  notation  —  the  linear  vector 
function,  the  dyad,  and  the  dyadic  with  their  appropriate 
symbolization. 

The  simplest  example  of  a  linear  vector  function  is  the 
product  of  a  scalar  constant  and  a  vector.  The  vector  r' 

r'  =  cr  (1) 

is  a  linear  function  of  r.  A  more  general  linear  function 
may  be  obtained  by  considering  the  components  of  r  individ- 
ually. Let  i,  j,  k  be  a  system  of  axes.  The  components  of 
r  are 

i  •  r,      j  •  r,      k  •  r. 

Let  each  of  these  be  multiplied  by  a  scalar  constant  which 
may  be  different  for  the  different  components. 

Cji.r,        caj.r,        c3  k  •  r. 


LINEAR    VECTOR  FUNCTIONS  261 

Take  these  as  the  components  of  a  new  vector  r' 

r'  =  i    (CiiTj+j    (c2j.r)  +  k    (c8  k  •  r).       (2) 

The  vector  r'  is  then  a  linear  function  of  r.  Its  components 
are  always  equal  to  the  corresponding  components  of  r  each 
multiplied  by  a  definite  scalar  constant. 

Such  a  linear  function  has  numerous  applications  in  geom- 
etry and  physics.  If,  for  instance,  i,  j,  k  be  the  axes  of  a 
homogeneous  strain  and  cv  c2,  c3,  the  elongations  along  these 
axes,  a  point 

r=ri#+J7/  +  kz 

becomes  r'  =  i  cx  x  -f  j  c2  y  +  k  cs  2, 

or  r  '  =  i  cl  i  •  r  +  j  <?2  j  •  r  +  k  c3  k  •  r. 

This  sort  of  linear  function  occurs  in  the  theory  of  elasticity 
and  in  hydrodynamics.  In  the  theory  of  electricity  and 
magnetism,  the  electric  force  E  is  a  linear  function  of  the 
electric  displacement  D  in  a  dielectric.  For  isotropic  bodies 
the  function  becomes  merely  a  constant 


But  in  case  the  body  be  non-isotropic,  the  components  of  the 
force  along  the  different  axes  will  be  multiplied  by  different 
constants  kv  kv  k3.  Thus 

E  =  ik1i»'D  +  j&2  j  .D  +  k&gk.D. 

The  linear  vector  function  is  indispensable  in  dealing  with 
the  phenomena  of  electricity,  magnetism,  and  optics  in  non- 
isotropic  bodies. 

98.]  It  is  possible  to  define  a  linear  vector  function,  as  has 
been  done  above,  by  means  of  the  components  of  a  vector. 
The  most  general  definition  would  be 


262  VECTOR  ANALYSIS 

Definition :  A  vector  r'  is  said  to  be  a  linear  vector  func- 
tion of  another  vector  r  when  the  components  of  r'  along 
three  non-coplanar  vectors  are  expressible  linearly  with  scalar 
coefficients  in  terms  of  the  components  of  r  along  those  same 
vectors. 

If  r  =  #a  +  yb  +  zc,  where  [abc]  -£  0, 

and  r'  =  x' &  +  y'b  +  z'c, 

and  if  x'  =  a1x  +  b±y  +  c^z* 

(3) 


then  r'  is  a  linear  function  of  r.  (The  constants  a15  Jj,  cv 
etc.,  have  no  connection  with  the  components  of  a,  b,  c  par- 
allel to  i,  j,  k.)  Another  definition  however  is  found  to  be 
more  convenient  and  from  it  the  foregoing  may  be  deduced. 
Definition  :  A  continuous  vector  function  of  a  vector  is 
said  to  be  a  linear  vector  function  when  the  function  of  the 
sum  of  any  two  vectors  is  the  sum  of  the  functions  of  those 
vectors.  That  is,  the  function  /is  linear  if 


)•  (4) 

Theorem  :  If  a  be  any  positive  or  negative  scalar  and  if  / 
be  a  linear  function,  then  the  function  of  a  times  r  is  a  times 
the  function  of  r. 

/(ar)  =  a/(r).  (5) 

And  hence 

/Oi*i  +  «2r2  +  a3r3+  •••) 

=  «i/(*i)  +  «2/(r2)+  «8/(r8)  +  -  .  -         (5)' 

The  proof  of  this  theorem  which  appears  more  or  less 
obvious  is  a  trifle  long.  It  depends  upon  making  repeated 
use  of  relation  (4). 


LINEAR    VECTOR  FUNCTIONS  263 

Hence  /(2r)  =  2/(r). 

In  like  manner  /  (nr~)  =  w/(r) 

where  n  is  any  positive  integer. 

Let  m  be  any  other  positive  integer.     Then  by  the  relation 
just  obtained 


and 

Hence 


That  is,  equation  (5)  has  been  proved  in  case  the  constant  a 
is  a  rational  positive  number. 

To  show  the  relation  for  negative  numbers  note  that 

/(0)=/(0  +  0)=-2/(0). 
Hence  /(O)  =  0. 

But  /(O)  =/(r-r)  =/(r +(-r))  =/(r)  +/(-r). 

Hence  /(r)=-/(-r). 

To  prove  (5)  for  incommensurable  values  of  the  constant 
a,  it  becomes  necessary  to  make  use  of  the  continuity  of  the 
function  /.  That  is 

LIM     -  x     N  _ 
x=  a  J  ^     '  ~~ 

Let  x  approach  the  incommensurable  number  a  by  passing 
through  a  suite  of  commensurable  values.     Then 


Hence  LlM   /  (x  r)  =  a  f  (r) 

CC  _.  CL 


264  VECTOR  ANALYSIS 

Lm    (ar)=ar. 

/>>  — i-   fm     ^  * 

•&  tt 

Hence  /(ar)  =  a/(r) 

which  proves  the  theorem. 

Theorem :  A  linear  vector  function  /  (r)  is  entirely  deter- 
mined when  its  value  for  three  non-coplanar  vectors  a,  b,  c  are 
known. 

Let  l=/(a), 

m=/(b), 

•*=/(«)• 
Since  r  is  any  vector  whatsoever,  it    may  be  expressed  as 


Hence  /  (r)  =  x  1  +  y  m  +  z~ft 

99.]     In  Art.  97  a  particular  case  of  a  linear  function  was 
expressed  as 

r'  =  i  Cj  i .  r  +  j  c2  j  •  r  +  k  c3  k  .  r. 

For  the  sake  of  brevity  and  to  save  repeating  the  vector  r 
which  occurs  in  each  of  these  terms  in  the  same  way  this 
may  be  written  in  the  symbolic  form 

r/  =  (ic1i  +  jc2j  +  kc3k).r. 

In  like  manner  if  ap  a2,  a3  •  •  •  be  any  given  vectors,  and  br  b2, 
b3,  •  •  •  another  set  equal  in  number,  the  expression 

r'  =  aj  bj  «r  +  a2b2-r  +  a3b3  •  r  +  •  ••  (6) 

is  a  linear  vector  function  of  r ;  for  owing  to  the  distributive 
character  of  the  scalar  product  this  function  of  r  satisfies 
relation  (4).  For  the  sake  of  brevity  r'  may  be  written  sym- 
bolically in  the  form 

r'  =  (ax  bj  +  aa  b2  +  a3  ba  +  •••)'  r.          (6)' 


LINEAR    VECTOR  FUNCTIONS  265 

No  particular  physical  or  geometrical  significance  is  to  be 
attributed  at  present  to  the  expression 

(a1b1  +  aaba  +  a,ba  +  ...).  (7) 

It  should  be  regarded  as  an  operator  or  symbol  which  con- 
verts the  vector  r  into  the  vector  r'  and  which  merely 
affords  a  convenient  and  quick  way  of  writing  the  relation 
(6). 

Definition :  An  expression  a  b  formed  by  the  juxtaposition 
of  two  vectors  without  the  intervention  of  a  dot  or  a  cross  is 
called  a  dyad.  The  symbolic  sum  of  two  dyads  is  called  a 
dyadic  binomial ;  of  three,  a  dyadic  trinomial ;  of  any  num- 
ber, a  dyadic  polynomial.  For  the  sake  of  brevity  dyadic 
binomials,  trinomials,  and  polynomials  will  be  called  simply 
dyadics.  The  first  vector  in  a  dyad  is  called  the  antecedent ; 
and  the  second  vector,  the  consequent.  The  antecedents  of  a 
dyadic  are  the  vectors  which  are  the  antecedents  of  the 
individual  dyads  of  which  the  dyadic  is  composed.  In  like 
manner  the  consequents  of  a  dyadic  are  the  consequents  of 
the  individual  dyads.  Thus  in  the  dyadic  (7)  a-p  a2,  a3  •  •  •  are 
the  antecedents  and  bx,  b2,  b3  •  •  •  the  consequents. 

Dyadics  will  be  represented  symbolically  by  the  capital 
Greek  letters.  When  only  one  dyadic  is  present  the  letter 
0  will  generally  be  used.  In  case  several  are  under  consid- 
eration other  Greek  capitals  will  be  employed  also.  With 
this  notation  (7)  becomes 

0  =  a1b1  +  aaba  +  a8b8+...,  (7)' 

and  (6)'  may  now  be  written  briefly  in  the  form 

r'  =  0  .  r.  (8) 

By  definition      0  •  r  =  &l  bx  •  r  -f  a2  b2  •  r  +  a3  b3  •  r  +  •  •  • 

The  symbol  <P»r  is  read  0  dot  r.  It  is  called  the  direct 
product  of  0  into  r  because  the  consequents  bx,  b2,  b3  •  •  •  are 


266  VECTOR  ANALYSIS 

multiplied  into  r  by  direct  or  scalar  multiplication.  The 
order  of  the  factors  0  and  r  is  important.  The  direct 
product  of  r  into  0  is 

r.  <P  =  r-(si1l)1  +  a2b2  +  a3b3  +  •••)    ' 

=  r.a1b1  +  r.a2b2  +  r»a3b3  +  ...  (9) 

Evidently  the  vectors  ^»  r_andjr^_i?  are  in  general  different. 

Definition :  When  the  dyadic  0  is  multiplied  into  r  as  0  •  r, 
0  is  said  to  be  a  pref  actor  to  r.  When  r  is  multiplied  in  0  as 
r  •  0,  0  is  said  to  be  a  post/ actor  to  r. 

A  dyadic  0  used  either  as  a  pref  actor  or  as  a  postf actor  to  a 
vector  r  determines  a  linear  vector  function  of  r.  The  two  linear 
vector  functions  thus  obtained  are  in  general  different  from 
one  another.  They  are  called  conjugate  linear  vector  func- 
tions. The  two  dyadics 

<P  =  a1b1  +  aab2  +  a3b8  +  ••• 
and  #-  W  =  bj  ^  +  b2  a2  +  b3  a3  +  •  •  • » 

each  of  which  may  be  obtained  from  the  other  by  inter- 
changing the  antecedents  and  consequents,  are  called  conju- 
gate dyadics.  The  fact  that  one  dyadic  is  the  conjugate  of 
another  is  denoted  by  affixing  a  subscript  C  to  either. 

Thus  ¥  =  0a    .    0=¥c. 

Theorem:  A  dyadic  used  as  a  postf  actor  gives  the  same 
result  as  its  conjugate  used  as  a  prefactor.  That  is 

r  •  0  =  0C .  r.  (9) 

100.]  Definition :  Any  two  dyadics  0  and  W  are  said  to 
be  equal 

when  0  .  r  =  W  •  r        for  all  values  of  r, 

or  when          r  .  0  =  r  .  W        for  all  values  of  r,  (10) 

or  when      B  •  0  .  r  =  s  •  W  •  r   for  all  values  of  s  and  r. 


LINEAR   VECTOR  FUNCTIONS  267 

The  third  relation  is  equivalent  to  the  first.  For,  if  the 
vectors  0  •  r  and  ¥  •  r  are  equal,  the  scalar  products  of  any 
vector  &  into  them  must  be  equal.  And  conversely  if  the 
scalar  product  of  any  and  every  vector  s  into  the  vectors  0  •  r 
and  ¥•!  are  equal,  then  those  vectors  must  be  equal.  In 
like  manner  it  may  be  shown  that  the  third  relation  is  equiva- 
lent to  the  second.  Hence  all  three  are  equivalent. 

Theorem  :  A  dyadic  0  is  completely  determined  when  the 

values  *..,   <z>.b,    </>.„, 

where  a,  b,  c  are  any  three  non-coplanar  vectors,  are  known. 
This  follows  immediately  from  the  fact  that  a  dyadic  defines 
a  linear  vector  function.     If 


consequently  two  dyadics  0  and  W  are  equal  provided  equa- 
tions (10)  hold  for  three  non-coplanar  vectors  r  and  three 
non-coplanar  vectors  s. 

Theorem  :  Any  linear  vector  function  /  may  be  represented 
by  a  dyadic  0  to  be  used  as  a  prefactor  and  by  a  dyadic  ¥, 
which  is  the  conjugate  of  0,  to  be  used  as  a  postfactor. 

The  linear  vector  function  is  completely  determined  when 
its  values  for  three  non-coplanar  vectors  (say  i,  j,  k)  are 
known  (page  264).  Let 


Then  the  linear  function  /  is  equivalent  to  the  dyadic   0 

givenby  0  =  ai  +  bj  +  ck, 

to  be  used  as  a  postfactor  ;  and  to  the  dyadic  ¥ 

¥=  00  =  ia  +  jb  +  kc, 
to  be  used  as  a  prefactor. 

/  (r)  =  0  •  r  =  r  •  0^ 


268  VECTOR  ANALYSIS 

The  study  of  linear  vector  functions  therefore  is  identical 
with  the  study  of  dyadics. 

Definition  :  A  dyad  a  b  is  said  to  be  multiplied  by  a  scalar 
a  when  the  antecedent  or  the  consequent  is  multiplied  by 
that  scalar,  or  when  a  is  distributed  in  any  manner  between 
the  antecedent  and  the  consequent.  If  a  =  a'  a" 

a  (a  b)  =  (a  a)  b  =  a  (a  b)  =  (a'  a)  (a"  b). 

A  dyadic  0  is  said  to  be  multiplied  by  the  scalar  a  when 
each  of  its  dyads  is  multiplied  by  that  scalar.  The  product 

is  written 

a  0     or     0a. 

The  dyadic  a  0  applied  to  a  vector  r  either  as  a  prefactor  or 
as  a  postfactor  yields  a  vector  equal  to  a  times  the  vector 
obtained  by  applying  0  to  r  —  that  is 

(a  0)  .  r  =  a($.r). 

V 

Theorem  :  The  combination  of  vectors  in  a  dyad  is  distrib- 

utive.    That  is 


and  a  (b  +  c)  = 


This  follows  immediately  from  the  definition  of  equality  of 
dyadics  (10).     For 

[(a  +  b)  c]  •  r  =  (a  +  b)  c  •  r  =  a  c  •  r  -f  b  c  •  r  =  (a  c  +  b  c)  •  r 

and 

[a(b  +  c)J  «r  =  a(b  +  c)  .r  =  ab«r  +  ac»r  = 


Hence  it  follows  that  a  dyad  which  consists  of  two  factors, 
each  of  which  is  the  sum  of  a  number  of  vectors,  may  be 
multiplied  out  according  to  the  law  of  ordinary  algebra 
—  except  that  the  order  of  the  factors  in  the  dyads  must  le 
maintained. 


LINEAR    VECTOR   FUNCTIONS  269 


...     (11)' 
cl  +  cm  +  cn+  ... 


The  dyad  therefore  appears  as  a  product  of  the  two  vectors  of 
which  it  is  composed,  inasmuch  as  it  obeys  the  characteris- 
tic law  of  products  —  the  distributive  law.  This  is  a  justifi- 
cation for  writing  a  dyad  with  the  antecedent  and  conse- 
quent in  juxtaposition  as  is  customary  in  the  case  of  products 
in  ordinary  algebra. 

The  Nonion  Form  of  a  Dyadic 

lOL]     From  the  three  unit  vectors  i,  j,  k  nine  dyads  may 
be  obtained  by  combining  two  at  a  time.     These  are 

ii,      ij,      ik, 

ji>      jj,      jk,  (12) 

ki,       kj,       kk. 

If  all  the  antecedents  and  consequents  in  a  dyadic  0  be  ex- 
pressed in  terms  of  i,  j,  k,  and  if  the  resulting  expression  be 
simplified  by  performing  the  multiplications  according  to  the 
distributive  law  (11)'  and  if  the  terms  be  collected,  the  dyadic 
0  may  be  reduced  to  the  sum  of  nine  dyads  each  of  which  is 
a  scalar  multiple  of  one  of  the  nine  fundamental  dyads  given 

above. 

0  —  anii  +  a12  ij  +«13ik 

+  aal  j  i  +  a22  j  j  +  «23  i  k  (13) 

+  a31ki  +  a82kj  +  a33kk. 

This  is  called  the  nonion  form  of  0. 

Theorem  :  The  necessary  and  sufficient  condition  that  two 
dyadics  d>  and  ¥  be  equal  is  that,  when  expressed  in  nonion 


270  VECTOR  ANALYSIS 

form,  the  scalar  coefficients  of  the  corresponding  (?ads  be 
equal. 

If  the  coefficients  be  equal,  then  obviously 


for  any  value  of  r  and  the  dyadics  by  (10)  must  be 
Conversely,  if  the  dyadics  0  and  W  are  equal,  then  by 

s  •  0  •  r  =  s  •  ¥  •  r 

for  all  values  of  s  and  r.     Let  s  and  r  each  take  on  the  vlues 

i,j,k.     Then  14) 

i.  #.i  =  i.  gr.i      i»  0  »    =  i»  ?F.        i-<Z>.k  =  i.£.k 


k.  <P.i  =  k.  ^.i,    k«  <Z>.j  =  k.  r.j,    k.^.k^k.^k. 

But  these  quantities  are  precisely  the  nine  coefficients  in  ;he 
expansion  of  the  dyadics  0  and  W.  Hence  the  corresponding 
coefficients  are  equal  and  the  theorem  is  proved.1  Thu 
analytic  statement  of  the  equality  of  two  dyadics  can  some- 
times be  used  to  greater  advantage  than  the  more  fundamental 
definition  (10)  based  upon  the  conception  of  the  dyadic  as 
defining  a  linear  vector  function. 

Theorem  :  A  dyadic  0  may  be  expressed  as  the  sum  of  nine 
dyads  of  which  the  antecedents  are  any  three  given  non- 
coplanar  vectors,  a,  b,  c  and  the  consequents  any  three  given 
non-coplanar  vectors  1,  m,  n. 

Every  antecedent  may  be  expressed  in  terms  of  a,  b,  c  ; 
and  every  consequent,  in  terms  of  1,  m,  n.  The  dyadic  may 
then  be  reduced  to  the  form 


+  «21  bl  +  a23  bm  +  «23  bn  (15) 

+  a318l+  a32  cm  +  a33  en. 

9^ 

1  As  a  corollary  of  the  theorem  it  is  evident  that  the  nine  dyads  (12)  are  in- 
dependent.   None  of  them  may  be  expressed  linearly  in  terms  of  the  others. 


LINEAR    VECTOR  FUNCTIONS  271 

This  expression  of  0  is  more  general  than  that  given  in 
(13).  It  reduces  to  that  expression  when  each  set  of  vectors 
a,  b,  c  and  1,  m,  n  coincides  with  i,  j,  k. 

Theorem :  Any  dyadic  0  may  be  reduced  to  the  sum  of 
three  dyads  of  which  either  the  antecedents  or  the  consequents, 
but  not  both,  may  be  arbitrarily  chosen  provided  they  be  non- 
coplanar. 

Let  it  be  required  to  express  (P  as  the  sum  of  three  dyads 
of  which  a,  b,  c  are  the  consequents.  Let  1,  m,  n  be  any  other 
three  non-coplanar  vectors.  ®  may  then  be  expressed  as  in 
(15).  Hence 

0  =  a  (au  1  +  ala  m  +  als  n)  +  b  (a21 1  +  «22  m  +  a23  n) 

+  c  (a31 1  +  a32  m  +  a32  n), 

or  0  =  aA  +  bB  +  cC.  (16) 

In  like  manner  if  -it  be  required  to  express  0  as  the  sum  of 
three  dyads  of  which  the  three  non-coplanar  vectors  1,  m,  n  are 
the  consequents 

0  =  Ll  +  Mm  +  Nn,  (16)' 

where  L  =  au  a  +  azl  b  +  a31  c, 

M  =  al2  a  +  aZ2  b  +  «32  c, 
N  =  «18  a  +  «23  b  +  «33  c. 

The  expressions  (15),  (16),  (16)'  for  0  are  unique.  Two  equal 
dyadics  which  have  the  same  three  non-coplanar  ante- 
cedents, a,  b,  c,  have  the  same  consequents  A,  B,  C  —  these 
however  need  not  be  non-coplanar.  And  two  equal  dyadics 
which  have  the  same  three  non-coplanar  consequents  1,  m,  n, 
have  the  same  three  antecedents. 

102.]  Definition:  The  symbolic  product  formed  by  the  juxta- 
position of  two  vectors  a,  b  without  the  intervention  of  a  dot 
or  a  cross  is  called  the  indeterminate  product  of  the  two  vectors 
a  and  b. 


272  VECTOR  ANALYSIS 

The  reason  for  the  term  indeterminate  is  this.  The  two 
products  a  •  b  and  a  x  b  have  definite  meanings.  One  is  a 
certain  scalar,  the  other  a  certain  vector.  On  the  other  hand 
the  product  ab  is  neither  vector  nor  scalar  —  it  is  purely 
symbolic  and  acquires  a  determinate  physical  meaning  only 
when  used  as  an  operator.  The  product  a  b  does  not  obey 
the  commutative  law.  It  does  however  obey  the  distributive 
law  (11)  and  the  associative  law  as  far  as  scalar  multiplication 
is  concerned  (Art  100). 

TJieorem :  The  indeterminate  product  a  b  of  two  vectors  is 
the  most  general  product  in  which  scalar  multiplication  is 
associative. 

The  most  general  product  conceivable  ought  to  have  the 
property  that  when  the  product  is  known  the  two  factors  are 
also  known.  Certainly  no  product  could  be  more  general. 
Inasmuch  as  scalar  multiplication  is  to  be  associative,  that  is 

a(ab)  =  (aa)  b  =  a  (ab)  =  (a'a)  (a"b), 

it  will  be  impossible  to  completely  determine  the  vectors  a 
and  b  when  their  product  a  b  is  given.  Any  scalar  factor 
may  be  transferred  from  one  vector  to  the  other.  Apart  from 
this  possible  transference  of  a  scalar  factor,  the  vectors  com- 
posing the  product  are  known  when  the  product  is  known.  In 
other  words  — 

Theorem  :  If  the  two  indeterminate  products  a  b  and  a'  b' 
are  equal,  the  vectors  a  and  a',  b  and  b'  must  be  collinear  and 
the  product  of  the  lengths  of  a  and  b  (taking  into  account  the 
positive  or  negative  sign  according  as  a  and  b  have  respec- 
tively equal  or  opposite  directions  to  a'  and  b')  is  equal  to  the 
product  of  the  lengths  of  a'  and  b'. 

Let  a  =  al  i  +  aa  j  +  as  k, 


LINEAR    VECTOR  FUNCTIONS  273 


»•'*• 


Then  ab  =  «1&1 


+  «3&1  kj  +  «3Z>2    kj  +  a36 
and  a'V  =  «/&/  ii  +  a,'fta'  ij  +  a/Jg'  ik 

+  «2'V  ji+a2'&2'  jj  +  a2&3  jk 
+  as'&1'  ki  +  a3'62'  kj  +  a3'63'  kk. 

Since  ab  =  a'b'  corresponding  coefficients  are  equal.     Hence 

•i  s  •«  s  •§«s«i':  •!':«/! 

which  shows  that  the  vectors  a  and  a'  are  collinear. 

And  &i:&,:68  =  &i':W. 

which  shows  that  the  vectors  b  and  b'  are  collinear. 

But  al  b1  =  a^  &/. 

This  shows  that  the  product  of  the  lengths  (including  sign) 
are  equal  and  the  theorem  is  proved. 

The  proof  may  be  carried  out  geometrically  as  follows. 
Since  ab  is  equal  to  a'b' 

ab  •  r  =  a'b'  •  r 

for  all  values  of  r.  Let  r  be  perpendicular  to  b.  Then  b  •  r 
vanishes  and  consequently  b'«r  also  vanishes.  This  is  true 
for  any  vector  r  in  the  plane  perpendicular  to  b.  Hence  b  and 
b'  are  perpendicular  to  the  same  plane  and  are  collinear.  In 
like  manner  by  using  a  b  as  a  postf  actor  a  and  a'  are  seen 
to  be  parallel.  Also 

ab  •  b  =  a'b'  •  b, 

which  shows  that  the  products  of  the  lengths  are  the  same. 

18 


274  VECTOR  ANALYSIS 

The  indeterminate  product  a  b  imposes  Jive  conditions  upon 
the  vectors  a  and  b.  The  directions  of  a  and  b  are  fixed  and 
likewise  the  product  of  their  lengths.  The  scalar  product 
a  •  b,  being  a  scalar  quantity,  imposes  only  one  condition  upon 
a  and  b.  The  vector  product  a  x  b,  being  a  vector  quantity, 
imposes  three  conditions.  The  normal  to  the  plane  of  a  and 
b  is  fixed  and  also  the  area  of  the  parallelogram  of  which  they 
are  the  side.  The  nine  indeterminate  products  (12)  of  i,  j,  k 
into  themselves  are  independent.  The  nine  scalar  products 
are  not  independent.  Only  two  of  them  are  different. 


and        i  •  j  =  j  •  i  =  j  •  k  =  k«j=k»i  =  i»k  =  0. 

The  nine  vector  products  are  not  independent  either  ;  for 

ixi  =  jxj  =  kxk=:0, 

and     i  x  j  =  —  j  x  i,    j  x  k  =  —  k  x  j,     k  x  i  =  —  i  x  k. 

The  two  products  a  •  b  and  a  x  b  obtained  respectively  from 
the  indeterminate  product  by  inserting  a  dot  and  a  cross  be- 
tween the  factors  are  functions  of  the  indeterminate  product. 
That  is  to  say,  when  ab  is  given,  a  •  b  and  a  X  b  are  determined. 
For  these  products  depend  solely  upon  the  directions  of  a  and  b 
and  upon  the  product  of  the  length  of  a  and  b,  all  of  which 
are  known  when  ab  is  known.  That  is 

if     ab  =  a'b',      a  •  b  =  a'  •  b'  and  a  x  b  =  a'  x  b'.     (17) 

It  does  not  hold  conversely  that  if  a  •  b  and  a  x  b  are  known 
a  b  is  fixed  ;  for  taken  together  a  •  b  and  a  X  b  impose  upon  the 
vectors  only  four  conditions,  whereas  a  b  imposes  five.  Hence 
a  b  appears  not  only  as  the  most  general  product  but  as  the 
most  fundamental  product.  The  others  are  merely  functions 
of  it.  Their  functional  nature  is  brought  out  clearly  by  the 
notation  of  the  dot  and  the  cross. 


LINEAR    VECTOR  FUNCTIONS  275 

Definition :  A  scalar  known  as  the  scalar  of  0  may  be  ob- 
tained by  inserting  a  dot  between  the  antecedent  and  conse- 
quent of  each  dyad  in  a  dyadic.  This  scalar  will  be  denoted 
by  a  subscript  $  attached  to  0.  * 

If  0  =  a1b1  +  a2b2  +  a3b3  +  ... 

0S  =  ax .  bx  +  a2  •  b2  +  a3  .  b3  +  . . .          (18) 

In  like  manner  a  vector  known  as  the  vector  of  0  may  be 
obtained  by  inserting  a  cross  between  the  antecedent  and  con- 
sequent of  each  dyad  in  0.  This  vector  will  be  denoted  by 
attaching  a  subscript  cross  to  0. 

0X  =  at  x  bj  +  a2  x  b2  +  a3  x  b3  +  •  •  •      (19) 
If  0  be  expanded  in  nonion  form  in  terms  of  i,  j,  k, 

0s  =  an  +  aZ2  +  a^  (20) 

<Px  =  023  -  ^32)  *  +  («si  -  ais)  3  +  (ai2  ~  a2i)  k-  (21) 
Or  0S  =  i.  0-i  +j.0.j  +  k«  0-k,  (20)' 

<px  =  (j  .  0 .  k  -  k  •  0  •  j)  i  +  (k  •  0 •  i  -  i •  0 •  k)  j 

+  (i.0.j-J-0'i)k-  (21)' 

In  equations  (20)  and  (21)  the  scalar  and  vector  of  0  are 
expressed  in  terms  of  the  coefficients  of  0  when  expanded 
in  the  nonion  form.  Hence  if  0  and  ¥  are  two  equal 
dyadics,  the  scalar  of  0  is  equal  to  the  scalar  of  ¥  and  the 
vector  of  0  is  equal  to  the  vector  of  ¥. 

If  0  =  ¥,     0S  =  ¥3    and  0*  =  ¥x.  (22) 

From  this  it  appears  that  0S  and  0X  are  functions  of  0 
uniquely  determined  when  0  is  given.  They  may  sometimes 
be  obtained  more  conveniently  from  (20)  and  (21)  than  from 
(18)  and  (19),  and  sometimes  not. 

1  A  subscript  dot  might  be  used  for  the  scalar  of  *  if  it  were  sufficiently  distinct 
and  free  from  liability  to  misinterpretation. 


276  VECTOR  ANALYSIS 

Products  of  Dyadics 

103.]  In  giving  the  definitions  and  proving  the  theorems 
concerning  products  of  dyadics,  the  dyad  is  made  the  under- 
lying principle.  What  is  true  for  the  dyad  is  true  for  the 
dyadic  in  general  owing  to  the  fact  that  dyads  and  dyadics 
obey  the  distributive  law  of  multiplication. 

Definition:  The  direct  product  of  the   dyad  ab  into  the 

dyad  c  d  is  written  ,   ,,     ,    ,. 

(a  b)  •  (c  d) 

and  is  by  definition  equal  to  the  dyad  (b  •  c)  a  A 

(ab).(cd)  =  a(b.c)d  =  b-c  a4. 1  (23) 

That  is,  the  antecedent  of  the  first  and  the  consequent  of  the 
second  dyad  are  taken  for  the  antecedent  and  consequent 
respectively  of  the  product  and  the  whole  is  multiplied  by 
the  scalar  product  of  the  consequent  of  the  first  and  the 
antecedent  of  the  second. 

Thus  the  two  vectors  which  stand  together  in  the  product 

(a  b)  •  (c  d) 

are  multiplied  as  they  stand.  The  other  two  are  left  to  form 
a  new  dyad.  The  direct  product  of  two  dyadics  may  be 
defined  as  the  formal  expansion  (according  to  the  distributive 
law)  of  the  product  into  a  sum  of  products  of  dyads.  Thus 

^=(a1b1  +a2b2  +a3b3  +  ...) 
and  ^=(01*1  +c2d2  +  c3  d3  +  •  •  •) 

<P.  W=(^l^l  +a2b2  +a3b3  +  •••)• 

(cjdj  +c2d2  +c3d3  +  •••) 
=  a1b1»c1d1  4-a1b1-c2d2  +  a1b1  •  C3d3  +  •  •  • 
+  a2b2.c1d1  +a2b2.c2d2  +  a2b2.c3d3  +  ...    (23)' 
+  a3ba.c1d1  +a3b3»c2d2  +  a3b3  •  c8d3  +  •  •  • 
+ 

1  The  parentheses  may  he  omitted  in  each  of  these  three  expressions. 


LINEAR    VECTOR  FUNCTIONS  277 

0 .  W  =  bx .  cl  axdj  +b1»c2  axd2  +b1«c3  a1d3  +  --- 

+  t>2  '  °1    a2dl    X  1>2'C2    &2d2    +  1>2  '  C3    *2  d3  +  '  '  ' 

+  b3  •  Cj  a3  dx  +  b3  .  c2  a3  d2  +  b3 .  c3  a3d3  +  •  •  • 
+ (23)" 

The  product  of  two  dyadics  0  and  W  is  a  dyadic  0  •  W. 

Theorem :  The  product  0  •  W  of  two  dyadics  0  and  W  when 
regarded  as  an  operator  to  be  used  as  a  prefactor  is  equiva- 
lent to  the  operator  W  followed  by  the  operator  0. 

Let  Q  =  0  •  W. 

To  show  Q  •  r  =  0  •  (  ¥ .  r), 

or  (<Z>.  ?F).r  =  </>.(r.r).  (24) 

Let  ab  be  any  dyad  of  0  and  c  d  any  dyad  of  ¥. 

(a  b  •  c  d)  •  r  =  b'c  (a  d  •  r)  =  (b  •  c)  (d  •  r)  a, 
a  b  •  (c  d  •  r)  =  a  b  •  c  (d  •  r)  =  (b  •  c)  (d  •  r)  a, 
Hence  (a  b  •  c  d)  •  r  =  a  b  •  (c  d  •  r). 

The  theorem  is  true  for  dyads.  Consequently  by  virtue  of 
the  distributive  law  it  holds  true  for  dyadics  in  general. 

If  r  denote  the  position  vector  drawn  from  an  assumed  origin 
to  a  point  P  in  space,  r'  =  W  •  r  will  be  the  position  vector  of 
another  point  P\  and  r"  =  $•(¥••*)  will  be  the  position 
vector  of  a  third  point  P".  That  is  to  say,  W  defines  a  trans- 
formation of  space  such  that  the  points  P  go  over  into  the 
points  P'.  0  defines  a  transformation  of  space  such  that  the 
points  P'  go  over  into  the  points  P".  Hence  ¥  followed  by 
0  carries  P  into  P".  The  single  operation  0  •  ¥  also  carries 
P  into  P". 

Theorem:  Direct  multiplication  of  dyadics  obeys  the  dis- 
tributive law.  That  is 


278  VECTOR  ANALYSIS 

&•(¥  +  ¥'}  =  <P.  ¥  +  0-  ¥' 
and  ($'  +  <P)  •  ¥  =  &'  •  ¥  +  0  .  ¥.  (25) 

Hence  in  general  the  product 

(0+  <p>  +  0"  +...).  (¥+  ¥'+  ¥"+..•) 

may  be  expanded  formally  according  to  the  distributive  law. 

Theorem  :  The  product  of  three  dyadics  <P,  ¥,  Q  is  associa- 
tive,    That  is        (^.^.Q=0.(¥.^  (26) 

and  consequently  either  product  may  be  written  without 
parentheses,  as  $.?.&  (26)' 

The  proof  consists  in  the  demonstration  of  the  theorem  for 
three  dyads  ab,  cd,  ef  taken  respectively  from  the  three 
dyadics  0,  ¥,  Q. 

(ab.cd)  »ef  =  (b«c)  ad*  ef  =  (b»c)  (d«  e)  af, 
ab.(cd.ef)  =  (d-e)  ab»cf  =  (d»e)  (b»c)  af. 

The  proof  may  also  be  given  by  considering  <#,  ¥,  and  Q 
as  operators 

{(<?  •  V)  .  Q\  .  r  =  (0  .  W)  •  (Q  .  r). 

Let  £.r  =  r' 

{($•¥)•£}.*  =  (&.  ¥).i'=<l>.  (¥•!'). 
Let  ¥  .  r'  =  r", 

\(0.  W)*Q\  .r=  0.i"  =  T'". 
Again  {(?.(?T.  £)}  .  r  =  <^>.  [(?T.  J2)  .r]. 


Hence  {((^.  ?F)  .  j2|  .r  =  {#.  (?F.         .r 

for  all  values  of  r.    Consequently 


LINEAR    VECTOR  FUNCTIONS  279 

The  theorem  may  be  extended  by  mathematical  induction 
to  the  case  of  any  number  of  dyadics.  The  direct  product 
of  any  number  of  dyadics  is  associative.  Parentheses  may 
be  inserted  or  omitted  at  pleasure  without  altering  the  result. 

It  was  shown  above  (24)  that 

(0.  T).r  =  &.(¥.T)=  0.  W*r.  (24)' 

Hence  the  product  of  two  dyadics  and  a  vector  is  associative. 
The  theorem  is  true  in  case  the  vector  precedes  the  dyadics 
and  also  when  the  number  of  dyadics  is  greater  than  two. 
But  the  theorem  is  untrue  when  the  vector  occurs  between 
the  dyadics.  The  product  of  a  dyadic,  a  vector,  and  another 
dyadic  is  not  associative. 

(0.r).  W  *  <P.(r.  ¥).  (27) 

Let  a  b  be  a  dyad  of  <P,  and  c  d  a  dyad  of  W. 

(a  b  •  r)  •  c  d  =  b  •  r  (a  •  c  d)  =  (b  •  r)  (a  •  c)  d, 
ab  •  (r  •  c  d)  =  ab  •  d  (r  •  c)  =  b  •  d  (r  •  c)  a 
Hence  (ab«r)  •  cd  ±  ab»  (r»cd). 

The  results  of  this  article  may  be  summed  up  as  follows : 

Theorem:  The  direct  product  of  any  number  of  dyadics 
or  of  any  number  of  dyadics  with  a  vector  factor  at  either 
end  or  at  both  ends  obeys  the  distributive  and  associative 
laws  of  multiplication  —  parentheses  may  be  inserted  or 
omitted  at  pleasure.  But  the  direct  product  of  any  number 
of  dyadics  with  a  vector  factor  at  some  other  position  than  at 
either  end  is  not  associative  —  parentheses  are  necessary  to 
give  the  expression  a  definite  meaning. 

Later  it  will  be  seen  that  by  making  use  of  the  conjugate 
dyadics  a  vector  factor  which  occurs  between  other  dyadics 
may  be  placed  at  the  end  and  hence  the  product  may  be 
made  to  assume  a  form  in  which  it  is  associative. 


280  VECTOR  ANALYSIS 

104.]  Definition:  The  skew  products  of  a  dyad  ab  into 
a  vector  r  and  of  a  vector  r  into  a  dyad  ab  are  denned 
respectively  by  the  equations 

(ab)  x  r  =  a(b  x  r), 
rx(ab)  =  (r  x  a)b. 

The  skew  product  of  a  dyad  and  a  vector  at  either  end  is  a 
dyad.     The  obvious  extension  to  dyadics  is 

=  a1b1  X  r  +  a2b2  xr  +  a3b3  x  r  H 

=  r  x  ajbj  +  r  x  a2b2  +  r  x  asbg  +  ... 

Theorem:  The  direct  product  of  any  number  of  dyadics 
multiplied  at  either  end  or  at  both  ends  by  a  vector  whether 
the  multiplication  be  performed  with  a  cross  or  a  dot  is 
associative.  But  in  case  the  vector  occurs  at  any  other 
position  than  the  end  the  product  is  not  associative.  That  is, 

(rx  $)  •  ?F  =  rx(0«?r)  =  rx  <!>•¥, 

(r  x  0).s  =  r  x  (0-s)  =r  x  <P»s,  (29) 

r  .  (0  x  s)  =  (r  •  0)  x  s  =  r  •  0  x  s, 
r  x  (0  x  s)  =  (r  x  0)  x  s  =  r  x  <P  x  a, 
but  ¥.(TX  0)  *  (?T.r)x  V. 

Furthermore  the  expressions 

s  •  r  x  0  and  0  x  r  •  8 
can  have  no  other  meaning  than 

s  •  r  x  0  =  s  •  (r  x  $), 
d>  x  r  •  s  =  (0  x  r)  •  s, 


LINEAR    VECTOR  FUNCTIONS  281 

since  the  product  of  a  dyadic  0  with  a  cross  into  a  scalar  s  •  r 
is  meaningless.  Moreover  since  the  dot  and  the  cross  may 
be  interchanged  in  the  scalar  triple  product  of  three  vectors 
it  appears  that 

s  •  r  x  d>  =  (B  x  r)  •  <P, 

0  x  r  •  s  =  0  .  (r  x  s),  (31) 

and  0  •  (r  x  W)  =  (0  x  r)  .  ¥. 

The  parentheses  in  these  expressions  cannot  be  omitted 
without  incurring  ambiguity. 

$.(r  X  s)  *  (<Z>.r)  x  s, 
(s  x  r)  •  0  *  &  x  (r  •  0),  (31)' 

(0  •  r)  x  W  #  0  x  (r  .  ¥). 

The  formal  skew  product  of  two  dyads  a  b  and  c  d  would  be 
(ab)  x  (c  d)  =  a  (b  x  c)  d. 

In  this  expression  three  vectors  a,  b  x  c,  d  are  placed  side 
by  side  with  no  sign  of  multiplication  uniting  them.  Such 

an  expression 

rst  (32) 

is  called  a  triad ;  and  a  sum  of  such  expressions,  a  triadic. 
The  theory  of  triadics  is  intimately  connected  with  the  theory 
of  linear  dyadic  functions  of  a  vector,  just  as  the  theory  of 
dyadics  is  connected  with  the  theory  of  linear  vector  functions 
of  a  vector.  In  a  similar  manner  by  going  a  step  higher 
tetrads  and  tetradics  may  be  formed,  and  finally  polyads  and 
polyadics.  But  the  theory  of  these  higher  combinations  of 
vectors  will  not  be  taken  up  in  this  book.  The  dyadic 
furnishes  about  as  great  a  generality  as  is  ever  called  for  in 
practical  applications  of  vector  methods. 


282  VECTOR  ANALYSIS 

Degrees  of  Nullity  of  Dyadics 

105.]  It  was  shown  (Art.  101)  that  a  dyadic  could  always 
be  reduced  to  a  sum  of  three  terms  at  most,  and  this  reduction 
can  be  accomplished  in  only  one  way  when  the  antecedents 
or  the  consequents  are  specified.  In  particular  cases  it  may 
be  possible  to  reduce  the  dyadic  further  to  a  sum  of  two 
terms  or  to  a  single  term  or  to  zero.  Thus  let 

0  =  al  +  bm  +  cn. 

If  1,  m,  n  are  coplanar  one  of  the  three  may  be  expressed 
in  terms  of  the  other  two  as 

1  =  x  m  +  y  n. 

Then  (P^azm  +  ayn  +  bm  +  cn, 

<P  =  (a  x  4-  b)  m  +  (a  y  +  c)  n. 

The  dyadic  has  been  reduced  to  two  terms.  If  1,  m,  n  were 
all  collinear  the  dyadic  would  reduce  to  a  single  term  and  if 
they  all  vanished  the  dyadic  would  vanish. 

Theorem :  If  a  dyadic  0  be  expressed  as  the  sum  of  three 
terms 

d>  =  al  +  bm  +  en 

of  which  the  antecedents  a,  b,  c  are  known  to  be  non-coplanar, 
then  the  dyadic  may  be  reduced  to  the  sum  of  two  dyads 
when  and  only  when  the  consequents  are  coplanar. 

The  proof  of  the  first  part  of  the  theorem  has  just  been 
given.  To  prove  the  second  part  suppose  that  the  dyadic 
could  be  reduced  to  a  sum  of  two  terms 

<P  =  dp  +  eq 

and  that  the  consequents  1,  m,  n  of  0  were  non-coplanar. 
This  supposition  leads  to  a  contradiction.  For  let  1',  m',  n' 
be  the  system  reciprocal  to  1,  m,  n.  That  is, 

,  _  mx  n         ,  _  n  x  1         ,  _  1  x  m  g 
"  [Imn]'         ~jTmn]'    n  =  [Tmn]  ' 


LINEAR   VECTOR    FUNCTIONS  283 

The  vectors  1',  m',  n'  exist  and  are  non-coplanar  because 
1,  m,  n  have  been  assumed  to  be  non-coplanar.  Any  vector  r 
may  be  expressed  in  terms  of  them  as 

r  =  x\'  +  ym'  +  zn,' 

<p.r  =  (al  +  bin  +  en)  •  (x\*  +  ym'  +  zn'). 
But  1  •  1'  =  m  •  m'  =  n  •  n'  =  1, 

and  1  •  m'  =  1'  •  m  =  m  •  n'  =  m'  •  n  =  n  •  1'  =  n'  •  1  =  0. 
Hence  0»r  =  xa  +  yb  +  zc. 

By  giving  to  r  a  suitable  value  the  vector  0  •  r  may  be  made 
equal  to  any  vector  in  space. 

But         0  «  r  =  (dp  +  e  q)  •  r  =  d  (p  •  r)  +  e  (q  •  r). 

This  shows  that  0  •  r  must  be  coplanar  with  d  and  e.  Hence 
0  •  r  can  take  on  only  those  vector  values  which  lie  in  the 
plane  of  d  and  e.  Thus  the  assumption  that  1,  m,  n  are  non- 
coplanar  leads  to  a  contradiction.  Hence  1,  m,  n  must  be 
coplanar  and  the  theorem  is  proved. 

Theorem  :  If  a  dyadic  0  be  expressed  as  the  sum  of  three 
terms 

0  =  &l  +  bm  +  en, 

of  which  the  antecedents  a,  b,  c  are  known  to  be  non-coplanar, 
the  dyadic  0  can  be  reduced  to  a  single  dyad  when  and  only 
when  the  consequents  1,  m,  n  are  collinear. 

The  proof  of  the  first  part  was  given  above.     To  prove 
the  second  part  suppose  0  could  be  expressed  as 


Let  W—0  x  p  =  dp  x  p  =  dO  =  0, 

?F=al  xp  +  bmxp  +  cnxp. 


284  VECTOR  ANALYSIS 

From  the  second  equation  it  is  evident  that  W  used  as  a 
postfactor  for  any  vector 

r  =  x  a;  +  y  b'  +  20', 

where  a',  b',  c'  is  the  reciprocal  system  to  a,  b,  c  gives 
r.  ?r=iclxp  +  2/inxp  +  2nxp. 

From  the  first  expression 

r  .  0  =  0. 
Hence  zlxp+ymxp  +  znxp 

must  be  zero  for  every  value  of  r,  that  is,  for  every  value  of  #, 
y,  2.  Hence 

lxp  =  0,        m  x  p  =  0,        n  x  p  =  0. 

Hence  1,  m,  and  n  are  all  parallel  to  p  and  the  theorem  has 
been  demonstrated. 

If  the  three  consequents  1,  m,  n  had  been  known  to  be  non- 
coplanar  instead  of  the  three  antecedents,  the  statement  of 
the  theorems  would  have  to  be  altered  by  interchanging  the 
words  antecedent  and  consequent  throughout.  There  is  a  fur- 
ther theorem  dealing  with  the  case  in  which  both  antecedents 
and  consequents  of  0  are  coplanar.  Then  0  is  reducible  to 
the  sum  of  two  dyads. 

106.]  Definition:  A  dyadic  which  cannot  be  reduced  to 
the  sum  of  fewer  than  three  dyads  is  said  to  be  complete.  A 
dyadic  which  may  be  reduced  to  the  sum  of  two  dyads,  but 
cannot  be  reduced  to  a  single  dyad  is  said  to  be  planar.  In 
case  the  plane  of  the  antecedents  and  the  plane  of  the  con- 
sequents coincide  when  the  dyadic  is  expressed  as  the  sum  of 
two  dyads,  the  dyadic  is  said  to  be  uniplanar.  A  dyadic 
which  may  be  reduced  to  a  single  dyad  is  said  to  be  linear. 
In  case  the  antecedent  and  consequent  of  that  dyad  are  col- 


LINEAR   VECTOR  FUNCTIONS  285 

linear,  the  dyadic  is  said  to  be  unilinear.  If  a  dyadic  may  be 
so  expressed  that  all  of  its  terms  vanish  the  dyadic  is  said  to 
be  zero.  In  this  case  the  nine  coefficients  of  the  dyadic  as 
expressed  in  nonion  form  must  vanish. 

The  properties  of  complete,  planar,  uniplanar,  linear,  and 
unilinear  dyadics  when  regarded  as  operators  are  as  follows. 
Let 

s  =  0  .  r  and  t  =  r  •  0. 

If  ¥  is  complete  s  and  t  may  be  made  to  take  on  any  desired 
value  by  giving  r  a  suitable  value. 


As  0  is  complete  1,  m,  n  are  non-coplanar  and  hence  have  a 
reciprocal  system  1',  m',  n'. 

B  =  0  •  (xlf  +  ym'  +  za')  =  ica  +  yb  +  20. 

In  like  manner  a,  b,  c  possess  a  system  of  reciprocals  a',  b',  c'. 

t  =  (xa.1  +  y\>'  +  zc')  •  0  =  x\  +  y  m  +  zn. 

A  complete  dyadic  0  applied  to  a  vector  r  cannot  give  zero 
unless  the  vector  r  itself  is  zero. 

If  0  is  planar  the  vector  s  may  take  on  any  value  in  the  plane 
of  the  antecedents  and  t  any  value  in  the  plane  of  the  consequents 
of  0  ;  but  no  values  out  of  those  planes.  The  dyadic  0  when 
used  as  a  prefactor  reduces  every  vector  r  in  space  to  a  vector 
in  the  plane  of  the  antecedents.  In  particular  any  vector  r 
perpendicular  to  the  plane  of  the  consequents  of  0  is  reduced 
to  zero.  The  dyadic  0  used  as  a  postfactor  reduces  every 
vector  r  in  space  to  a  vector  in  the  plane  of  the  consequents 
of  0.  In  particular  a  vector  perpendicular  to  the  plane  of 
the  antecedents  of  0  is  reduced  to  zero.  In  case  the  dyadic 
is  uniplanar  the  same  statements  hold. 

If  0  is  linear  the  vector  s  may  take  on  any  value  collinear 
with  the  antecedent  of  0  and  t  any  value  collinear  with  the  con- 


286  VECTOR  ANALYSIS 

sequent  of  <P  ;  but  no  other  values.  The  dyadic  <P  used  as  a 
prefactor  reduces  any  vector  r  to  the  line  of  the  antecedent 
of  0.  In  particular  any  vectors  perpendicular  to  the  con- 
sequent of  0  are  reduced  to  zero.  The  dyadic  0  used  as  a 
postfactor  reduces  any  vector  r  to  the  line  of  the  consequent 
of  0.  In  particular  any  vectors  perpendicular  to  the  ante- 
cedent of  0  are  thus  reduced  to  zero. 

If  $  is  a  zero  dyadic  the  vectors  s  and  t  are  loth  zero  no 
matter  what  the  value  of  r  may  be. 

Definition  :  A  planar  dyadic  is  said  to  possess  one  degree  of 
nullity.  A  linear  dyadic  is  said  to  possess  two  degrees  of 
nullity.  A  zero  dyadic  is  said  to  possess  three  degrees  of  nul- 
lity or  complete  nullity. 

107.]  Theorem  :  The  direct  product  of  two  complete  dyadics 
is  complete;  of  a  complete  dyadic  and  a  planar  dyadic, 
planar  ;  of  a  complete  dyadic  and  a  linear  dyadic,  linear. 

Theorem  :  The  product  of  two  planar  dyadics  is  planar 
except  when  the  plane  of  the  consequent  of  the  first  dyadic 
in  the  product  is  perpendicular  to  the  plane  of  the  antece- 
dent of  the  second  dyadic.  In  this  case  the  product  reduces 
to  a  linear  dyadic  —  and  only  in  this  case. 

Let  0  =  &'b    +  ab 


2, 


Q  =  0  .  W. 

The  vector  B  =  ¥  •  r  takes  on  all  values  in  the  plane  of 
and  c0 


The  vector  B'  =  0  •  s  takes  on  the  values 

s'  =  0  .  s  =  x  (bj  .  c:)  ax  +  y  (bj  •  C2)  at 

+  X  0>2  '  Cl)   a2  +  2^  (b2  '  C)  a2> 

B'  =  J*  (bj  -oj)  +  y  (bj  •  c2)}  ax  +  {x  (b2  •  Cl)  +  y  (b,  •  c2)}: 


LINEAR   VECTOR  FUNCTIONS  287 

Let  s'  =  JT'E!  +  y'aa, 

where  x'  =  x  (bx  •  c^  +  #  (bj  •  c2), 

and  y'  =  a?  (b2  •  cx)  +  y  (ba  •  c2). 

These  equations  may  always  be  solved  for  x  and  y  when 
any  desired  values  x'  and  y'  are  given  —  that  is,  when  s'  has 
any  desired  value  in  the  plane  of  a^  and  a2  —  unless  the 
determinant 

b2  •  Cj    b2  •  c, 

But  by  (25),  Chap.  II.,  this  is  merely  the  product 
(b!  x  b,)  •  (c,  X  Co)  =  0. 

x.     j.  a  /  \     j.  &  / 

The  vector  bj  x  b2  is  perpendicular  to  the  plane  of  the  con- 
sequents of  0 ;  and  cx  X  c2,  to  the  plane  of  the  antecedents  of 
¥.  Their  scalar  product  vanishes  when  and  only  when  the 
vectors  are  perpendicular  —  that  is,  when  the  planes  are  per- 
pendicular. Consequently  Bf  may  take  on  any  value  in  the 
plane  of  ax  and  a2  and  0  •  ¥  is  therefore  a  planar  dyadic 
unless  the  planes  of  bx  and  b2,  Cj  and  c2  are  perpendicular. 
If  however  bj  and  b2,  Cj  and  c2  are  perpendicular  s'  can  take 
on  only  values  in  a  certain  line  of  the  plane  of  ax  and  a^,  and 
hence  (P  •  W  is  linear.  The  theorem  is  therefore  proved. 

Tlieorem  :  The  product  of  two  linear  dyadics  is  linear 
except  when  the  consequent  of  the  first  factor  is  perpen- 
dicular to  the  antecedent  of  the  second.  In  this  case  the 
product  is  zero  —  and  only  in  this  case. 

Theorem  :  The  product  of  a  planar  dyadic  into  a  linear  is 
linear  except  when  the  plane  of  the  consequents  of  the 
planar  dyadic  is  perpendicular  to  the  antecedent  of  the  linear 
dyadic.  In  this  case  the  product  is  zero  —  and  only  in  this 
case. 

Theorem:  The  product  of  a  linear  dyadic  into  a  planar 
dyadic  is  linear  except  when  the  consequent  of  the  linear 


288  VECTOR  ANALYSIS 

dyadic  is  perpendicular  to  the  plane  of  the  antecedents  of 
the  planar  dyadic.  In  this  case  the  product  is  zero  —  and 
only  in  this  case. 

It  is  immediately  evident  that  in  the  cases  mentioned  the 
products  do  reduce  to  zero.  It  is  not  quite  so  apparent  that 
they  can  reduce  to  zero  in  only  those  cases.  The  proofs  are 
similar  to  the  one  given  above  in  the  case  of  two  planar 
dyadics.  They  are  left  to  the  reader.  The  proof  of  the 
first  theorem  stated,  page  286,  is  also  left  to  the  reader. 

The  Idemf  actor;1  Reciprocals  and  Conjugates  of  Dyadics 

108.]  Definition  :  If  a  dyadic  applied  as  a  pref  actor  or  as 
a  postfactor  to  any  vector  always  yields  that  vector  the 
dyadic  is  said  to  be  an  idem/  actor.  That  is 

if  d>  •  r  =  r  for  all  values  of  r, 

or  if  r  •  <P  =  r  for  all  values  of  r, 

then  0  is  an  idemfactor.     The  capital  I  is  used  as  the  sym- 
bol for  an  idemfactor.     The  idemfactor  is  a  complete  dyadic. 
For  there  can  be  no  direction  in  which  I  •  r  vanishes. 
Theorem  :  When  expressed  in  nonion  form  the  idemfactor  is 


(33) 

Hence  all  idemfactors  are  equal. 

To  prove  that  the  idemfactor  takes  the  form  (33)  it  is 
merely  necessary  to  apply  the  idemfactor  I  to  the  vectors 
i,  j,  k  respectively.  Let 

I  =  an  ii  +  a12  ij  +  «13  ik 
+  a21ji  +  «22jj  +  «23jk 


1  In  the  theory  of  dyadics  the  idemfactor  I  plays  a  role  analogous  to  unity  in 
ordinary  algebra.    The  notation  is  intended  to  suggest  this  analogy. 


LINEAR   VECTOR  FUNCTIONS  289 

I»i  =  an  i+  «21  j  +  «31  k. 
If  I .  i  =  i, 

au  =  1  and  «21  =  a31  =  0. 

In  like  manner  it  may  be  shown  that  all  the  coefficients 
vanish  except  an,  «22,  «33  all  of  which  are  unity.     Hence 

I  =  ii  +  jj  +  kk.  (33) 

Theorem :   The  direct  product  of  any  dyadic  and  the  idem- 
factor  is  that  dyadic.     That  is, 

d> .  I  =  0  and  I  •  &  =  d>. 
For  (0.1)  T  =  0.(lT)  =#T, 

no  matter  what  the  value  of  r  may  be.     Hence,  page  266, 

0  . 1  =  0. 

In  like  manner  it  may  be  shown  that  I  •  0  =  0. 

Theorem:  If  a',  b',  c'  and  a,  b,  c  be  two  reciprocal  systems 
of  vectors  the  expressions 

I  =  aa'  +  bb'  +  cc',  (34) 

I  =  a'a  +  b'b  +  c'c 

are  idemfactors. 

For  by  (30)  and  (31)  Chap.  II., 

r  =  r«aa;  +  r-bb'  +  r«cc', 
and  r  =  r»a'a  +  r*b'b  +  r«c'c. 

Hence  the  expressions  must  be  idemfactors  by  definition. 
Theorem :  Conversely  if  the  expression 

4>  =  al  +  bin  +  en 

is  an  idemfactor  1,  m,  n  must  be  the  reciprocal  system  of 

a,  b,  c. 

19 


290  VECTOR  ANALYSIS 

In  the  first  place  since  0  is  the  idemfactor,  it  is  a  complete 
dyadic.  Hence  the  antecedents  a,  b,  c  are  non-coplanar  and 
possess  a  set  of  reciprocals  a',  b',  c'.  Let 

r  =  a;  a'  +  y  V  +  «c'. 
By  hypothesis  r  •  0  =  r. 

Then         r  •  0  =  x\  +  ym  +  zn  =  #a'  +  yV  +  zc 

for  all  values  of  r,  that  is,  for  all  values  of  #,  y,  z.     Hence  the 
corresponding  coefficients  must  be  equal.     That  is, 

1  =  a',      m  =  b',      n  =  c'. 

Theorem :  If  0  and  ¥  be  any  two  dyadics,  and  if  the  product 
0  •  ¥  is  equal  to  the  idemfactor;1  then  the  product  ¥  •  0, 
when  the  factors  are  taken  in  the  reversed  order,  is  also 
equal  to  the  idemfactor. 

Let  0  •¥  =  !. 

To  show  ¥  •  0  =  I. 

r»(<P.  ¥)  =r.I  =  r, 

r»(0.  F)  •  0  =  f  *0, 

r  .  (0  .  ?T)  .  0  =  (r  •  0)  •  (  V  •  (?)  =  r  •  #. 

This  relation  holds  for  all  values  of  r.     As  0  is  complete  r  •  0 
must  take  on  all  desired  values.     Hence  by  definition 

¥  .  0  =  I. 

If  the  product  of  two  dyadics  is  an  idemfactor,  that  product 
may  be  taken  in  either  order. 

109.]  Definition:  When  two  dyadics  are  so  related  that 
their  product  is  equal  to  the  idemfactor,  they  are  said  to  be 

1  This  necessitates  both  the  dyadics  *  and  V  to  be  complete.  For  the  product 
of  two  incomplete  dyadics  is  incomplete  and  hence  could  not  be  equal  to  the 
idemfactor. 


LINEAR   VECTOR  FUNCTIONS  291 

reciprocals^     The  notation  used  for  reciprocals  in  ordinary 
algebra  is  employed  to  denote  reciprocal  dyadics.     That  is, 


=0-i  =      .     (35) 

Theorem:  Reciprocals  of  the  same  or  equal  dyadics  are 
equal. 

Let  0  and  ¥  be  two  given  equal  dyadics,  0~l  and  W~l 
their  reciprocals  as  defined  above.  By  hypothesis 

0=  ¥, 


and  ¥•  ¥~i  =  I. 

To  show  0-i  =  ¥~i. 

0.0-^  =  1=  W.  ¥~i. 
As  0=  W,       0.0-i=  0.  ¥~\ 

0-1  .  0  .  0-1  =0-1.0.   ¥~\ 
0-1.0  =  1, 

I.0-i  =  0-i  =  I.  ¥~i  =  ¥~i. 
Hence  0~i  =  ¥~\ 

The  reciprocal  of  0  is  the  dyadic  whose  antecedents  are  the 
reciprocal  system  to  the  consequents  of  0  and  whose  conse- 
quents are  the  reciprocal  system  to  the  antecedents  of  0. 

If  a  complete  dyadic  0  be  written  in  the  form 

<P  =  al  +  bm  +  en, 

its  reciprocal  is       0~i  =  !'&'  +  m'  b'  +  n'  c'.  (36) 

For     (al  +  bm  +  cn)  •  (l'a'  +  n'V  +  n'c')  =aa'  +  bV  +  cc'. 

Theorem  :  If  the  direct  products  of  a  complete  dyadic  0 
into  two  dyadics  ¥  and  Q  are  equal  as  dyadics  then  ¥  and  Q 

1  An  incomplete  dyadic  has  no  (finite)  reciprocal. 


292  VECTOR  ANALYSIS 

are  equal.  If  the  product  of  a  dyadic  0  into  two  vectors 
r  and  s  (whether  the  multiplication  be  performed  with  a  dot 
or  a  cross)  are  equal,  then  the  vectors  r  and  s  are  equal. 
That  is, 

if  O.  ¥  =  0*8,      then  ¥=  Q, 

and  if  &  •  r  =  0  •  s,       then  r  =  s,  (37) 

and  if  0  x  r  =  0  x  s,      then  r  =  s. 

This  may  be  seen  by  multiplying  each  of  the  equations 
through  by  the  reciprocal  of  0, 


i  .  0  .  r  =  r  =  &~l  •  0  •  s  =  s, 

x  r  =  I  x  r  =  0"1  •  <P  x  s  =  I  x  s. 


To  reduce  the  last  equation  proceed   as  follows.     Let  t  be 

any  vector, 

t-Ixr  =  t«Ixs, 

t  •  I  =  t. 
Hence  t  x  r  =  t  x  s. 

As  t  is  any  vector,  r  is  equal  to  s. 

Equations  (37)  give  what  is  equivalent  to  the  law  of  can- 
celation  for  complete  dyadics.  Complete  dyadics  may  be 
canceled  from  either  end  of  an  expression  just  as  if  they 
were  scalar  quantities.  The  cancelation  of  an  incomplete 
dyadic  is  not  admissible.  It  corresponds  to  the  cancelation 
of  a  zero  factor  in  ordinary  algebra. 

110.]  Theorem:  The  reciprocal  of  the  product  of  any 
number  of  dyadics  is  equal  to  the  product  of  the  reciprocals 
taken  in  the  opposite  order. 

It  will  be  sufficient  to  give  the  proof  for  the  case  in  which 
the  product  consists  of  two  dyadics.  To  show 


LINEAR   VECTOR  FUNCTIONS  293 

(0.  ST)-i  =  3F-i.  0-i, 

0  .  W  •  W~l  •  0-1  =  0 .  (  ¥  •  ¥~l)  .  0-i  =  0  .  0-1  =  I. 
Hence  (0  •  2T)  •  (  ST"1 .  0-1)  =  I. 

Hence        0  •  W  and  5F"1  •  0"1  must  be  reciprocals.     That  is, 
(0.  ?T)-i=  sr-i.  0-i. 

The  proof  for  any  number  of  dyadics  may  be  given  in  the 
same  manner  or  obtained  by  mathematical  induction. 

Definition :  The  products  of  a  dyadic  0,  taken  any  number 
of  times,  by  itself  are  called  powers  of  0  and  are  denoted  in 
the  customary  manner. 

0  .  0  =  02, 
0  .  0  .  0  =  0  .  02  =  03^ 

and  so  forth. 

TJieorem  :  The  reciprocal  of  a  power  of  0  is  the  power  of 
the  reciprocal  of  0. 

(0»)-i  =  (0-i)»  =  0-».  (37) 

The  proof  follows  immediately  as  a  corollary  of  the  preced- 
ing theorem.  The  symbol  0~"  may  be  interpreted  as  the 
nth  power  of  the  reciprocal  of  0  or  as  the  reciprocal  of 
the  nth  power  of  0. 

If  0  be  interpreted  as  an  operator  determining  a  trans- 
formation of  space,  the  positive  powers  of  0  correspond  to 
repetitions  of  the  transformation.  The  negative  powers  of  0 
correspond  to  the  inverse  transformations.  The  idemfactor 
corresponds  to  the  identical  transformation — that  is,  no  trans- 
formation at  all.  The  fractional  and  irrational  powers  of  0 
will  not  be  denned.  They  are  seldom  used  and  are  not 
single-valued.  For  instance  the  idemfactor  I  has  the  two 
square  roots  ±  I.  But  in  addition  to  these  it  has  a  doubly 
infinite  system  of  square  roots  of  the  form 

0  =  -ii  +      +  kk. 


294  VECTOR  ANALYSIS 

Geometrically  the  transformation 

r'=  <P-r 

is  a  reflection  of  space  in  the  j  k-plane.  This  transformation 
replaces  each  figure  by  a  symmetrical  figure,  symmetrically 
situated  upon  the  opposite  side  of  the  j  k-plane.  The  trans- 
formation is  sometimes  called  perversion.  The  idemfactor 
has  also  a  doubly  infinite  system  of  square  roots  of  the  form 


Geometrically  the  transformation 

r'  =  W  •  r 

is  a  reflection  in  thei-axis.  This  transformation  replaces  each 
figure  by  its  equal  rotated  about  the  i-axis  through  an  angle 
of  180°.  The  idemfactor  thus  possesses  not  only  two  square 
roots ;  but  in  addition  two  doubly  infinite  systems  of  square 
roots ;  and  it  will  be  seen  (Art.  129)  that  these  are  by  no 
means  all. 

111.]  The  conjugate  of  a  dyadic  has  been  defined  (Art.  99) 
as  the  dyadic  obtained  by  interchanging  the  antecedents  and 
consequents  of  a  given  dyadic  and  the  notation  of  a  subscript 
G  has  been  employed.  The  equation 

r  .  0  =  0C .  r  (9) 

has  been  demonstrated.  The  following  theorems  concerning 
conjugates  are  useful. 

Theorem :  The  conjugate  of  the  sum  or  difference  of  two 
dyadics  is  equal  to  the  sum  or  difference  of  the  conjugates, 

Theorem :  The  conjugate  of  a  product  of  dyadics  is  equal 
to  the  product  of  the  conjugates  taken  in  the  opposite  order. 


LINEAR   VECTOR  FUNCTIONS  295 

It  will  be  sufficient  to  demonstrate  the  theorem  in  case 
the  product  contains  two  factors.     To  show 

(0.T)C=VC.0C,  (40) 

(0.  8%.r  =  r-(0.  W)  =  (r.Q).  V, 


Hence  (0  •  ¥)c=  Wc  •  4>c> 

Theorem  :  The  conjugate  of  the  power  of  a  dyadic  is  the 
power  of  the  conjugate  of  the  dyadic. 

(*•%=  W  =  **  (41) 

This  is  a  corollary  of  the  foregoing  theorem.    The  expression 
0nc  may  be  interpreted  in  either  of  two  equal  ways. 

Theorem  :  The  conjugate  of  the  reciprocal  of  a  dyadic  is 
equal  to  the  reciprocal  of  the  conjugate  of  the  dyadic. 

(0-i)c=(00)-i  =  0^.  (42) 

For  (0-%  *<PC  =  (0  . 


The  idemfactor  is  its  own  conjugate  as  may  be  seen  from 
the  nonion  form. 

I  =  i  i  +  j  j  +  k  k 
(0c)-i.0c  =  I. 

Hence  ((P^"1  •  4>c  =  (  0~^0  •  ^ 

Hence  ((^c)-1  =  (0-% 

The  expression  &c~l  may  therefore  be  interpreted  in  either 
of  two  equivalent  ways  —  as  the  reciprocal  of  the  conjugate 
or  as  the  conjugate  of  the  reciprocal. 

Definition:     If  a  dyadic  is  equal  to  its  conjugate,  it  is  said 
to  be  self  -conjugate.     If  it  is  equal  to  the  negative  of  its  con- 


296  VECTOR  ANALYSIS 

jugate,  it  is  said  to  be  anti-self-conjugate.     For  se£/-conjugate 

dyadics. 

r  .  0  =  0  .  r,      0  =  0C. 

For  anti-self-conjugate  dyadics 

r  .  0  =  —  0.  r,      0  =  —  00. 

Theorem  :  Any  dyadic  may  be  divided  in  one  and  only  one 
way  into  two  parts  of  which  one  is  self-conjugate  and  the 
other  anti-self-conjugate. 


For  0=(0+0C}  +    (0-0C\  (43) 

But  (0  +  0c)c  =  0C  +  0CC  =  0C+  0, 

and  (0  -  0C-)C  =  0C-  0CC  =  0c-0. 

Hence  the  part  |(0  +  ^c)  ^s  self  -conjugate  ;  and  the  part 
|(0—  0C),  anti-self-conjugate.  Thus  the  division  has  been 
accomplished  in  one  way.  Let 

5  (0  +  <2>c)  =  *' 


0  =  0'  +  0". 

Suppose  it  were  possible  to  decompose  0  in  another  way 
into  a  self  -conjugate  and  an  anti-self-conjugate  part.  Let 
then 


Where      (0'  +  £)  =  (0'  +  J2)c  =  0'c  +  Qc  =  0'  +  ^ 

Hence  if  (0'  +  ,12)  is  self  -con  jugate,  Q  is  self-conjugate. 
-  (0"  -Q)=  (0"  _  £)c  =  0"c  _  J2e  =  _  0"  -  ^ 

Hence    if   (0"  —  £)  is    anti-self-conjugate    J2  is    anti-self- 
conjugate. 


LINEAR   VECTOR  FUNCTIONS  297 

Any  dyadic  which  is  both  self-conjugate  and  anti-self-conju- 
gate is  equal  to  its  negative  and  consequently  vanishes. 
Hence  Q  is  zero  and  the  division  of  0  into  two  parts  is 
unique. 

Anti-self-conjugate  Dyadics.      The  Vector  Product 
112.]  In  case  0  is  any  dyadic  the  expression 


gives  the  anti-self-conjugate  part  of  0.  If  0  should  be  en- 
tirely anti-self-conjugate  0  is  equal  to  0".  Let  therefore  0" 
be  any  anti-self-conjugate  dyadic, 

0"  =  1(0-0^. 

Suppose  0  =  al  +  bm  +  cn, 

0  —  0C  =  &1  —  la  +  bm  —  mb  +  cn  —  nc, 

<?"•  r  =  al»  r  —  la«r-f-bm»r  —  mb»r  +  cn«r  —  nc«r. 

But  al*  r  —  la«r  =  —  (a  x  1)  x  r, 

bm»r  —  mb»r  —  —  (bxm)xr, 

cn»r  —  nc»r  =  —  (cxn)xr. 

Hence      0"  .  r  =  —  ^(axl  +  bxm  +  cxn)xr. 
But  by  definition        <Px=:axl-|-bxm  +  cxii. 
Hence  0"  •  r  =  -  \  0*  X  r, 

r  .  0"  =  0"c  .  r  =  -  0"  •  r  =  \0K  x  r  =  -  \  r  x  0X. 

The  results  may  be  stated  in  a  theorem  as  follows. 

Theorem  :  The  direct  product  of  any  anti-self-conjugate 
dyadic  and  the  vector  r  is  equal  to  the  vector  product  of 
minus  one  half  the  vector  of  that  dyadic  and  the  vector  r. 


298  VECTOR  ANALYSIS 

3  (0-0*)  "  =  -3  <2>xX 


Theorem  :  Any  anti-self-conjugate  dyadic  $"  possesses  one 
degree  of  nullity.  It  is  a  uniplanar  dyadic  the  plane  of 
whose  consequents  and  antecedents  is  perpendicular  to  <PX", 
the  vector  of  0. 

This  theorem  follows  as  a  corollary  from  equations  (44). 

Theorem  :  Any  dyadic  0  may  be  broken  up  into  two  parts 
of  which  one  is  self-conjugate  and  the  other  equivalent  to 
minus  one  half  the  vector  of  <P  used  in  cross  multiplication. 

d>  .  r  =  0'  .  r  -  g  <PX  X  r, 

or  symbolically          0  •  =  0'  •  -  \  0X  X.  (45) 

113.]  Any  vector  c  used  in  vector  multiplication  defines  a 
linear  vector  function.  For 


Hence  it  must  be  possible  to  represent  the  operator  c  x  as  a 
dyadic.  This  dyadic  will  be  uniplanar  with  plane  of  its 
antecedents  and  consequents  perpendicular  to  c,  so  that  it 
will  reduce  all  vectors  parallel  to  c  to  zero.  The  dyadic  may 
be  found  as  follows 


By  (31)  I  .  (  c  x  I)  =  (I  x  c)  •  I, 

(I  x  o)  •  r  =  {  (I  x  o)  •  1}  •  r=  {I  •  (o  x  I)}  •  r 

=  I  •  (c  x  I)  .  r  =  (c  x  I)  •  r. 
Hence  c  x  r  —  (I  x  c)  •  r  =  (c  x  I)  •  r, 

and  r  x  c  =  r  •  (I  x  c)  =  r  •  (c  X  I).  (46) 

This  may  be  stated  in  words. 


LINEAR   VECTOR  FUNCTIONS  299 

Theorem :  The  vector  c  used  in  vector  multiplication  with 
a  vector  r  is  equal  to  the  dyadic  I  x  c  or  c  x  I  used  in  direct 
multiplication  with  r.  If  c  precedes  r  the  dyadics  are  to  be 
used  as  prefactors ;  if  c  follows  r,  as  postfactors.  The  dyadics 
I  X  c  and  c  x  I  are  anti-self-conjugate. 

In  case  the  vector  c  is  a  unit  vector  the  application  of  the 
operator  c  x  to  any  vector  r  in  a  plane  perpendicular  to  c  is 
equivalent  to  turning  r  through  a  positive  right  angle  about 
the  axis  c.  The  dyadic  c  X  I  or  I  x  c  where  c  is  a  unit  vector 
therefore  turns  any  vector  r  perpendicular  to  c  through  a 
right  angle  about  the  line  c  as  an  axis.  If  r  were  a  vector 
lying  out  of  a  plane  perpendicular  to  c  the  effect  of  the  dyadic 
I  X  c  or  c  x  I  would  be  to  annihilate  that  component  of  r  which 
is  parallel  to  c  and  turn  that  component  of  r  which  is  perpen- 
dicular to  c  through  a  right  angle  about  c  as  axis. 

If  the  dyadic  be  applied  twice  the  vectors  perpendicular  to 
r  are  rotated  through  two  right  angles.  They  are  reversed  in 
direction.  If  it  be  applied  three  times  they  are  turned  through 
three  right  angles.  Applying  the  operator  I  x  c  or  c  x  I  four 
times  brings  a  vector  perpendicular  to  c  back  to  its  original 
position.  The  powers  of  the  dyadic  are  therefore 

(I  x  c)2  =  (c  x  I)2  =  -  (I  -  c  c), 

(I  x  c)3  =  (c  x  I)3  =  -  I  x  c  =  -  c  X  I, 

(47) 
(I  x  c)4  =  (c  x  I)4  =  I  -  c  c, 

(I  x  c)5  =  (c  x  I)5  =  I  x  c  =  c  x  I. 

It  thus  appears  that  the  dyadic  I  x  c  or  c  x  I  obeys  the  same 
law  as  far  as  its  powers  are  concerned  as  the  scalar  imaginary 
V  —  1  in  algebra. 

The  dyadic  Ixc  or  cxlisa  quadrantal  versor  only  for 
vectors  perpendicular  to  c.  For  vectors  parallel  to  c  it  acts 
as  an  annihilator.  To  avoid  this  effect  and  obtain  a  true 


300  VECTOR  ANALYSIS 

quadrantal  versor  for  all  vectors  r  in  space  it  is  merely  neces- 
sary to  add  the  dyad  c  c  to  the  dyadic  I  x  c  or  c  X  I. 


If 


(48) 


The  dyadic  X  therefore  appears  as  a  fourth  root  of  the 
idemfactor.  The  quadrantal  versor  X  is  analogous  to  the 
imaginary  V  —  1  of  a  scalar  algebra.  The  dyadic  X  is  com- 
plete and  consists  of  two  parts  of  which  I  x  c  is  anti-self- 
conjugate  ;  and  c  c,  self  -conjugate. 

114.]     If  i,  j,  k  are  three  perpendicular  unit  vectors 

Ixi  =  ixl  =  kj—  jk, 

I  xj=j  x  I  =  ik-ki,  (49) 

I  x  k  =  k  x  I  =  j  i  —  i  j, 

as  may  be  seen  by  multiplying  the  idemfactor 

I  =  ii  +  jj  +  kk 

into  i,  j,  and  k  successively.  These  expressions  represent 
quadrantal  versors  about  the  axis  i,  j,  k  respectively  combined 
with  annihilators  along  those  axes.  They  are  equivalent, 
when  used  in  direct  multiplication,  to  i  x,  jx,  k  x  respectively, 

(IXk)»=(kxI)»  =  -(ii+jj), 


The  expression  (I  x  k)4  is  an  idemfactor  for  the  plane  of  i  and 
j,  but  an  annihilator  for  the  direction  k.  In  a  similar  man- 
ner the  dyad  k  k  is  an  idemfactor  for  the  direction  k,  but  an 


LINEAR   VECTOR  FUNCTIONS  301 

annihilator  for  the  plane  perpendicular  to  k.  These  partial 
idemfactors  are  frequently  useful. 

If  a,  b,  c  are  any  three  vectors  and  a',  V,  c'  the  reciprocal 

system, 

aa'  +  bb' 

used  as  a  prefactor  is  an  idemfactor  for  all  vectors  in  the 
plane  of  a  and  b,  but  an  annihilator  for  vectors  in  the  direc- 
tion c.  Used  as  a  postfactor  it  is  an  idemfactor  for  all  vectors, 
in  the  plane  of  a'  and  b',  but  an  annihilator  for  vectors  in  the 
direction  c'.  In  like  manner  the  expression 

cc' 

used  as  a  prefactor  is  an  idemfactor  for  vectors  in  the  direction 
c,  but  for  vectors  in  the  plane  of  a  and  b  it  is  an  annihilator. 
Used  as  a  postfactor  it  is  an  idemfactor  for  vectors  in  the 
direction  c',  but  an  annihilator  for  vectors  in  the  plane  of  af 
and  b',  that  is,  for  vectors  perpendicular  of  c. 
If  a  and  b  are  any  two  vectors 

(a  x  b)  x  I  =  I  x  (a  x  b)  =  b  a  -  ab.  (50) 

For 

{(a  x  b)  x  I}«r  =  (a  x  b)  x  r  =  ba«r  —  abT  =  (ba  —  ab)«r. 

The  vector  a  X  b  in  cross  multiplication  is  therefore  equal  to 
the  dyadic  (b  a  —  a  b)  in  direct  multiplication.  If  the  vector 
is  used  as  a  prefactor  the  dyadic  must  be  so  used. 

(a  xb)  x  r  =  (b  a  —  a  b)  •  r, 

r  x  (a  x  b)  =  r  •  (b  a  -  ab).  (51) 

This  is  a  symmetrical  and  easy  form  in  which  to  remember 
the  formula  for  expanding  a  triple  vector  product. 


302  VECTOR  ANALYSIS 

Reduction  of  Dyadics  to  Normal  Form 

115.]  Let  0  be  any  complete  dyadic  and  let  r  be  a  unit 
vector.  Then  the  vector  r' 

r'=  0-r 

is  a  linear  function  of  r.  When  r  takes  on  all  values  consis- 
tent with  its  being  a  unit  vector  —  that  is,  when  the  terminus 
of  r  describes  the  surface  of  a  unit  sphere,  —  the  vector  r' 
varies  continuously  and  its  terminus  describes  a  surface.  This 
surface  is  closed.  It  is  fact  an  ellipsoid.1 

Theorem :  It  is  always  possible  to  reduce  a  complete  dyadic 
to  a  sum  of  three  terms  of  which  the  antecedents  among 
themselves  and  the  consequents  among  themselves  are  mutu- 
ally perpendicular.  This  is  called  the  normal  form  of  0. 

<P  =  ai'i  +  Jj'j  +  ck'k. 
To  demonstrate  the  theorem  consider  the  surface  described 

by 

r'=0-r. 

As  this  is  a  closed  surface  there  must  be  some  direction  of  r 
which  makes  r'  a  maximum  or  at  any  rate  gives  r'  as  great 
a  value  as  it  is  possible  for  r'  to  take  on.  Let  this  direction 
of  r  be  called  i,  and  let  the  corresponding  direction  of  r'  — 
the  direction  in  which  r'  takes  on  a  value  at  least  as  great  as 
any  —  be  called  a.  Consider  next  all  the  values  of  r  which 
lie  in  a  plane  perpendicular  to  i.  The  corresponding  values 
of  r'  lie  in  a  plane  owing  to  a  fact  that  0  •  r  is  a  linear  vector 

1  This  may  be  proved  as  follows : 

*  =  *>*    r  =  *-L  r'srUe-i. 
Hence  r«r=l=    r'.  (*c~1'*  ~l)'i'  =  r'  •  V-r'. 

By  expressing  V  in  nonion  form,  the  equation  r'  •  V  •  r'  =  1  is  seen  to  be  of  the  second 
degree.  Hence  r'  describes  a  quadric  surface.  The  only  closed  quadric  surface 
is  the  ellipsoid. 


LINEAR   VECTOR  FUNCTIONS  303 

function.  Of  these  values  of  r'  one  must  be  at  least  as  great 
as  any  other.  Call  this  b  and  let  the  corresponding  direction 
of  r  be  called  j.  Finally  choose  k  perpendicular  to  i  and  j 
upon  the  positive  side  of  plane  of  i  and  j.  Let  c  be  the 
value  of  r'  which  corresponds  to  r  =  k.  Since  the  dyadic  0 
changes  i,  j,  k  into  a,  b,  c  it  may  be  expressed  in  the  form 

<P  =  &i  +  bj  +  ck. 

It  remains  to  show  that  the  vectors  a,  b,  c  as  determined 
above  are  mutually  perpendicular. 

r'  =  (ai  +  bj  +ck)»r, 

dr'  =  (ai  -f-  bj  +  ck)  .dr, 

r'.  dr'  =  r'  •  ai»  dr  +  r'  •  bj  •  dr  +  r'«  ck«  dr. 

When  r  is  parallel  to  i,  r'  is  a  maximum  and  hence  must  be 
perpendicular  to  dr'.  Since  r  is  a  unit  vector  dr  is  always 
perpendicular  to  r.  Hence  when  r  is  parallel  to  i 


If  further  dr  is  perpendicular  to  j,  r'«c  vanishes,  and  if 
dr  is  perpendicular  to  k,  r'«b  vanishes.  Hence  when  r  is 
parallel  to  i,  r  '  is  perpendicular  to  both  b  and  c.  But  when 
r  is  parallel  to  i,  r'  is  parallel  to  a.  Hence  a  is  perpendicular 
to  b  and  c.  Consider  next  the  plane  of  j  and  k  and  the 
plane  of  b  and  c.  Let  r  be  any  vector  in  the  plane  of  j  and  k. 

r'  =  (bj  +  ck)«r, 

dr'  =  (bj  +  ck)  •  dr, 

rr'dr'  =  r'''b  j  •  dr  +  r'  •  c   k  •  dr. 

When  r  takes  the  value  j,  r'  is  a  maximum  in  this  plane  and 
hence  is  perpendicular  to  dr'.  Since  r  is  a  unit  vector  it  is 


304  VECTOR  ANALYSIS 

perpendicular  to  dr.     Hence   when  r  is   parallel  to  j,  dr 
is  perpendicular  to  j,  and 


Hence  r'  •  c  is  zero.     But  when  r  is  parallel  to  j,  r'  takes  the 
value  b.     Consequently  b  is  perpendicular  to  c. 

It  has  therefore  been  shown  that  a  is  perpendicular  to  b  and 
c,  and  that  b  is  perpendicular  to  c.  Consequently  the  three 
antecedents  of  0  are  mutually  perpendicular.  They  may  be 
denoted  by  i',  j',  k'.  Then  the  dyadic  0  takes  the  form 

0  =  ai'i  +6j'j  +  ck'k,  (52) 

where  a,  J,  c  are  scalar  constants  positive  or  negative. 

116.]  Theorem:  The  complete  dyadic  0  may  always  be 
reduced  to  a  sum  of  three  dyads  whose  antecedents  and 
whose  consequents  form  a  right-handed  rectangular  system 
of  unit  vectors  and  whose  scalar  coefficients  are  either  all 
positive  or  all  negative. 

&  =  ±  (ai'i  +  &j'j  +  ck'k).  (53) 

The  proof  of  the  theorem  depends  upon  the  statements 
made  on  page  20  that  if  one  or  three  vectors  of  a  right-handed 
system  be  reversed  the  resulting  system  is  left-handed,  but 
if  two  be  reversed  the  system  remains  right-handed.  If  then 
one  of  the  coefficients  in  (52)  is  negative,  the  directions  of  the 
other  two  axes  may  be  reversed.  Then  all  the  coefficients 
are  negative.  If  two  of  the  coefficients  in  (52)  are  negative, 
the  directions  of  the  two  vectors  to  which  they  belong  may 
be  reversed  and  then  the  coefficients  in  0  are  all  positive. 
Hence  in  any  case  the  reduction  to  the  form  in  which  all 
the  coefficients  are  positive  or  all  are  negative  has  been 
performed. 

As  a  limiting  case  between  that  in  which  the  coefficients 
are  all  positive  and  that  in  which  they  are  all  negative  comes 


LINEAR   VECTOR  FUNCTIONS  305 

the  case  in  which  one  of  them  is  zero.  The  dyadic  then 
takes  the  form 

0  =  ai'i  -f  Jj'j  (54) 

and  is  planar.  The  coefficients  a  and  J  may  always  be  taken 
positive.  By  a  proof  similar  to  the  one  given  above  it  is 
possible  to  show  that  any  planar  dyadic  may  be  reduced  to 
this  form.  The  vectors  i'andj'are  perpendicular,  and  the 
vectors  i  and  j  are  likewise  perpendicular. 

It  might  be  added  that  in  case  the  three  coefficients  a,  6,  c 
in  the  reduction  (53)  are  all  different  the  reduction  can  be 
performed  in  only  one  way.  If  two  of  the  coefficients  (say 
a  and  &)  are  equal  the  reduction  may  be  accomplished  in  an 
infinite  number  of  ways  in  which  the  third  vector  k'  is  always 
the  same,  but  the  two  vectors  i',  j'  to  which  the  equal  coeffi- 
cients belong  may  be  any  two  vectors  in  the  plane  per- 
pendicular to  k.  In  all  these  reductions  the  three  scalar 
coefficients  will  have  the  same  values  as  in  any  one  of  them. 
If  the  three  coefficients  a,  &,  c  are  all  equal  when  0  is  reduced 
to  the  normal  form  (53),  the  reduction  may  be  accomplished 
in  a  doubly  infinite  number  of  ways.  The  three  vectors 
i',  j',  k'  may  be  any  right-handed  rectangular  system  in 
space.  In  all  of  these  reductions  the  three  scalar  coefficients 
are  the  same  as  in  any  one  of  them.  These  statements  will 
not  be  proved.  They  correspond  to  the  fact  that  the  ellipsoid 
which  is  the  locus  of  the  terminus  of  r'  may  have  three 
different  principal  axes  or  it  maybe  an  ellipsoid  of  revolution, 
or  finally  a  sphere. 

Theorem  :  Any  self  -con  jugate  dyadic  may  be  expressed  in 
the  form 


=  «ii  +  &jj  +  ckk  (55) 

where  a,  &,  and  c  are  scalars,  positive  or  negative. 

Let  <P  =  ai'i  +  fcj'j  +ck'k,  (52) 

6jj'  +  ckk', 


20 


306  VECTOR  ANALYSIS 

4>.<P0  =  a*i'i'  +  &2j7  +  c2k'k', 

<P0*0  =  a2ii  +  62jj  +  c2kk. 
Since  <P  =  <t>m 


j  +kk  =  i'i'  +j'j  +  k'k', 
-a2  1  =  (62-a2)  j'j'  +  (e2-a2)k'k', 


If  i  and  i'  were  not  parallel  (<P2  —  a2  I)  would  annihilate 
two  vectors  i  and  i'  and  hence  every  vector  in  their  plane. 
(02  —  a2  I)  would  therefore  possess  two  degrees  of  nullity 
and  be  linear.  But  it  is  apparent  that  if  a,  J,  c  are  different 
this  dyadic  is  not  linear.  It  is  planar.  Hence  i  and  i'  must 
be  parallel.  In  like  manner  it  may  be  shown  that  j  and  j  ', 
k  and  k'  are  parallel.  The  dyadic  0  therefore  takes  the  form 

0=  aii  +  Jjj  +  ckk 
where  a,  Z»,  c  are  positive  or  negative  scalar  constants. 

Double  Multiplication^ 

117.]  Definition  :  The  double  dot  product  of  two  dyads  is 
the  scalar  quantity  obtained  by  multiplying  the  scalar  product 
of  the  antecedents  by  the  scalar  product  of  the  consequents. 
The  product  is  denoted  by  inserting  two  dots  between  the 

ab:cd  =  a«c  b«d.  (56) 

This  product  evidently  obeys  the  commutative  law 
ab:cd  =  cd:ab, 

1  The  researches  of  Professor  Gibbs  upon  Double  Multiplication  are  here 
printed  for  the  first  time. 


LINEAR  VECTOR  FUNCTIONS  307 

and  the  distributive  law  both  with  regard  to  the  dyads  and 
with  regard  to  the  vectors  in  the  dyads.  The  double  dot 
product  of  two  dyadics  is  obtained  by  multiplying  the  prod- 
uct out  formally  according  to  the  distributive  law  into  the 
sum  of  a  number  of  double  dot  products  of  dyads. 

If  <P  =  a1b1  +  a2b2  +  a3b3  +  ... 

and  ¥  =  c1^.1  +  c2  d2  +  c3d3  +  •  •  • 


=  a1b1:c1d1  +  ajb^Oadg  +  a1b1:o8ds  +  ••• 

+  a2b2:c1d1  +  a2b2:c2d2  +  a2b2:c3d3  +  ...    (56)' 

+  agbgZCjdj  +  a3b3:c2d2  +  a3b3  :c3d3  +  .  •  . 


+     ...............      (56)" 

Definition:  The  double  cross  product  of  two  dyads  is  the 
dyad  of  which  the  antecedent  is  the  vector  product  of  the 
antecedents  of  the  two  dyads  and  of  which  the  consequent  is 
the  vector  product  of  the  consequent  of  the  two  dyads.  The 
product  is  denoted  by  inserting  two  crosses  between  the 

dyads 

ab^cd^axc    bxd.  (57) 

This  product  also  evidently  obeys  the  commutative  law 


308  VECTOR  ANALYSIS 

and  the  distributive  law  both  with  regard  to  the  dyads  and 
with  regard  to  the  vectors  of  which  the  dyads  are  composed. 
The  double  cross  product  of  two  dyadics  is  therefore  defined 
as  the  formal  expansion  of  the  product  according  to  the 
distributive  law  into  a  sum  of  double  cross  products  of 
dyads. 


a3b3 


1 

*>1 

y 

di  +  ajbjS 

C2 

d 

2 

+  *lbl 

X 
X 

c 

g  «•  +  ••' 

2 

ba 

£"! 

d1  +  a2b2; 

C2 

d. 

2 

+  aab2 

X 
X 

C 

,>  d-,  +  •  •  • 

a  Q 

3 

*3 

x  cl 

di  4-  a3b3  ^ 

c2 

sd 

2 

+  a3b3 

X 

C 

o  dq  +  •  •  • 

a  o  ' 

(57)' 


+  a3xc8  bgXdi  +a3xc2  bgXdj  +a3xc3  b3 

+    ...............     (57)" 

Theorem  :  The  double  dot  and  double  cross  products  of 
two  dyadics  obey  the  commutative  and  distributive  laws  of 
multiplication.  But  the  double  products  of  more  than  two 
dyadics  (whenever  they  have  any  meaning)  do  not  obey  the 
associative  law. 

0:  W=  W  10 

0  x  w  =  ¥  x  ^ 


The  theorem  is  sufficiently  evident  without  demonstration. 


LINEAR   VECTOR  FUNCTIONS  309 

Theorem  :  The  double  dot  product  of  two  fundamental 
dyads  is  equal  to  unity  or  to  zero  according  as  the  two 
dyads  are  equal  or  different. 


ij:ki  =  i»k  j  »i  =  0. 

Theorem:  The  double  cross  product  of  two  fundamental 
dyads  (12)  is  equal  to  zero  if  either  the  antecedents  or  the 
consequents  are  equal.  But  if  neither  antecedents  nor  con- 
sequents are  equal  the  product  is  equal  to  one  of  the  funda- 
mental dyads  taken  with  a  positive  or  a  negative  sign. 
That  is 

i  j  *  i  k  =  i  x  i    j  x  k  =  0 

ij*ki=ixk    jxi  =  +  jk. 

There  exists  a  scalar  triple  product  of  three  dyads  in 
which  the  multiplications  are  double.  Let  <P,  W,  Q  be  any 
three  dyadics.  The  expression 

0  *  WiQ 

is  a  scalar  quantity.  The  multiplication  with  the  double 
cross  must  be  performed  first.  This  product  is  entirely  in- 
dependent of  the  order  in  which  the  factors  are  arranged  or 
the  position  of  the  dot  and  crosses.  Let  ab,  cd,  and  ef  be 

three  dyads, 

ab  I  cd:ef  =  [ace]  [bdf].  (59) 

That  is,  the  product  of  three  dyads  united  by  a  double  cross 
and  a  double  dot  is  equal  to  the  product  of  the  scalar  triple 
product  of  the  three  antecedents  by  the  scalar  triple  product 
of  the  three  consequents.  From  this  the  statement  made 
above  follows.  For  if  the  dots  and  crosses  be  interchanged 
or  if  the  order  of  the  factors  be  permuted  cyclicly  the  two 
scalar  triple  products  are  not  altered.  If  the  cyclic  order  of 


310  VECTOR  ANALYSIS 

the  factors  is  reversed  each  scalar  triple  product  changes 
sign.     Their  product  therefore  is  not  altered. 

118.]    A  dyadic  0  may  be  multiplied  by  itself  with  double 

cross.    Let 

0  =  &l  +  bm  +  en 

0  $<P  =  (al  +  bm  +  en)  *  (al  +  bm  +  en) 

=  a  x  a  Ixl+axb  Ixm+axc  Ixn 
+  bxa  mxl  +  bxb  m  x  m  +  b  x  c  mxn 
+  cxa  nxl  +  cxb  nxm  +  cxc  nxn. 

The  products  in  the  main  diagonal  vanish.     The  others  are 
equal  in  pairs.     Hence 

<P*0  =  2  (bxc  mxn  +  cxa  nxl  +  axb  Ixm).  (60) 
If  a,  b,  c  and  1,  m,  n  are  non-coplanar  this  may  be  written 

0*0=-__A .(a'l'  +b'm'  +  c'n'>       (60)' 

[a be]  [Imn]  v 

The  product  0  $  0  is  a  species  of  power  of  0.  It  may  be  re- 
garded as  a  square  of  0  •  The  notation  02  will  be  employed 
to  represent  this  product  after  the  scalar  factor  2  has  been 
stricken  out. 

0*0 

0%  =  —%-  =  (b  x  c    mxn  +  cxa   nxl  +  axb   Ixm)  (61) 
3S 

The  triple  product  of  a  dyadic  0  expressed  as  the  sum  of 
three  dyads  with  itself  twice  repeated  is 

0$0:0  =  3  0^:0 

0Z:  0  =  (b  x  c    mxn  +  cxa    nxl  +  axb    Ixm) 
:  (al  +  bm  +  en). 

In  expanding  this  product  every  term  in  which  a  letter  is 
repeated  vanishes.    For  a  scalar  triple  product  of  three  vec- 


LINEAR  VECTOR   FUNCTIONS  311 

tors  two  of  which  are  equal  is  zero.     Hence  the  product 
reduces  to  three  terms  only 

02:<P=[bca]   [mnl]  +  [cab]  [nlm]  +  [abc]  [Imn] 
or  <P2:  0  =  3  [abc]  [Imn] 

<P*0:<Z>  =  6  [abc]  [Imn]. 

The  triple  product  of  a  dyadic  by  itself  twice  repeated  is 
equal  to  six  times  the  scalar  triple  product  of  its  antecedents 
multiplied  by  the  scalar  triple  product  of  its  consequents. 
The  product  is  a  species  of  cube.  It  will  be  denoted  by  03 
after  the  scalar  factor  6  has  been  stricken  out. 

d>  X0»0 
0.  =  ---  =  [abc]  [Imn].  (62) 


119.]  If  02  be  called  the  second  of  0  ;  and  03,  the  third  of 
0,  the  following  theorems  may  be  stated  concerning  the 
seconds  and  thirds  of  conjugates,  reciprocals,  and  products. 

Theorem  :  The  second  of  the  conjugate  of  a  dyadic  is  equal 
to  the  conjugate  of  the  second  of  that  dyadic.  The  third  of 
the  conjugate  is  equal  to  the  third  of  the  dyadic. 

<*•><=  W«  (63) 

4>,  =  (*<-),• 

Theorem:  The  second  and  third  of  the  reciprocal  of  a 
dyadic  are  equal  respectively  to  the  reciprocals  of  the  second 
and  third. 

<*-'),  =  (*,)-'=*,-* 

(<^-1)3  =  W1  =  <VJ 

Let  0  =  al 


>-i  =  V  a.'  +  m'b'+n'c'  (36) 


pi 
[abc] 


812  VECTOR  ANALYSIS 

(0a)-i  =  [abc]  [Imn]  (la  +  mb  +  nc) 

j  la  +  mb  +  nc 

;2~[a'b'c']   [1'm'n']' 

But        [a'Vc']  [abc]  =  1   and  [1'm'n']  [Imn]  =  1. 
Hence  (0,)'1  =  (0"1).  =  ®^- 


[abc]  [Imn] 

(0-i)3=[a'b'c']  [1'm'n']. 
Hence 


Theorem:  The  second  and  third  of  a  product  are  equal 
respectively  to  the  product  of  the  seconds  and  the  product  of 
the  thirds. 

(0-^2  =<V^2 
(*.*),«*.*,. 

Choose  any  three  non-coplanar  vectors  1,  m,  n  as  consequents 
of  0  and  let  1',  m',n'  be  the  antecedents  of  W. 

0  =  al  +  b  m  +  cn, 


(0.?P)2=bxc    exf  +  cxa    fxd  +  axb    dxe, 

cP2  =  bxcmxn  +  cxa    nxl  +  axb    Ixm, 
^2  =  m'  x  n'    e  x  f  +  n'  x  1'    f  x  d  +  1'  x  m'    dxe. 
Hence  ^2«?T2  =  bxc    exf  +  cxa    fxd  +  axb     dxe. 
Hence  (0  .  V)^  =  ^  •  Vz* 

((P.  F)3=  [abc]  [def] 


LINEAR   VECTOR   FUNCTIONS  313 

03=[abc]  [Imn], 

r3=[l'm'n']  [defj. 
Hence  <P3  ¥s  =  [a  b  c]  [d  e  f]. 

Hence  (0  •  r)3=  03  rs. 

Theorem  :  The  second  and  third  of  a  power  of  a  dyadic  are 
equal  respectively  to  the  powers  of  the  second  and  third  of 
the  dyadic. 

(0")2  =  W=02n 
(*")•  =  (*•)*=*."• 

Theorem  :  The  second  of  the  idemfactor  is  the  idemfactor. 
The  third  of  the  idemfactor  is  unity. 

I2  =  I 

(67) 
18  =  1. 

Theorem:  The  product  of  the  second  and  conjugate  of 
a  dyadic  is  equal  to  the  product  of  the  third  and  the 
idemfactor. 

0Z.4>C=0SI,  (68> 

<Z>2  =  b  x  c  mxn-fcxa  nxl  +  axb  Ixm, 

00  =  la  +  mb  +  nc, 
02  .  0C  =  [l  m  n]  (b  x  c  a  +  c  x  a  b  +  a  x  b  c). 

The  antecedents  a,  b,  c  of  the  dyadic  0  may  be  assumed  to 
be  non-coplanar.     Then 

(b  x  c   a  +  c  X  a  b  +  a  x  b  c)  =  [ab  c]  (a'a  +  V  b  +  c'  c) 

=  [abc]  I. 
Hence 


120.]  Let  a  dyadic  0  be  given.  Let  it  be  reduced  to  the 
sum  of  three  dyads  of  which  the  three  antecedents  are 
non-coplanar. 


314  VECTOR  ANALYSIS 

0  =  al  +  b  m  +  cn, 

<P2  =  b  x  c  mxn  +  cxa  nxl  +  axb  1  x  m, 
03  =  [abc]  [Imn]. 

Theorem:  The  necessary  and  sufficient  condition  that  a 
dyadic  0  be  complete  is  that  the  third  of  0  be  different  from 
zero. 

For  it  was  shown  (Art.  106)  that  both  the  antecedents  and 
the  consequents  of  a  complete  dyadic  are  non-coplanar. 
Hence  the  two  scalar  triple  products  which  occur  in  03 
cannot  vanish. 

Theorem:  The  necessary  and  sufficient  condition  that  a 
dyadic  0  be  planar  is  that  the  third  of  0  shall  vanish  but  the 
second  of  0  shall  not  vanish. 

It  was  shown  (Art.  106)  that  if  a  dyadic  0  be  planar  its  con- 
sequents 1,  m,  n  must  be  planar  and  conversely  if  the  conse- 
quents be  coplanar  the  dyadic  is  planar.  Hence  for  a  planar 
dyadic  03  must  vanish.  But  <P2  cannot  vanish.  Since  a, 
b,  c  have  been  assumed  non-coplanar,  the  vectors  b  x  c,  c  x  a, 
a  x  b  are  non-coplanar.  Hence  if  0%  vanishes  each  of  the 
vectors  mxn,  nxl,  Ixm  vanishes  —  that  is,  1,  m,  n  are  col- 
linear.  But  this  is  impossible  since  the  dyadic  0  is  planar 
and  not  linear. 

Theorem:  The  necessary  and  sufficient  condition  that  a 
non-vanishing  dyadic  be  linear  is  that  the  second  of  0,  and 
consequently  the  third  of  0,  vanishes. 

For  if  0  be  linear  the  consequents  1,  m,  n,  are  collinear. 
Hence  their  vector  products  vanish  and  the  consequents  of 
0Z  vanish.  If  conversely  0Z  vanishes,  each  of  its  consequents 
must  be  zero  and  hence  these  consequents  of  0  are  collinear. 

The  vanishing  of  the  third,  unaccompanied  by  the  vanish- 
ing of  the  second  of  a  dyadic,  implies  one  degree  of  nullity. 
The  vanishing  of  the  second  implies  two  degrees  of  nullity. 


LINEAR   VECTOR   FUNCTIONS  315 

The  vanishing  of  the  dyadic  itself  is  complete  nullity.  The 
results  may  be  put  in  tabular  form. 

03  ^  0,     0  is  complete. 
<P3  =  0,     02  :£  0,     0  is  planar.  (69) 

03  =  0,     </>2  =  0,     <P  *  0,     0  is  linear. 

It  follows  immediately  that  the  third  of  any  anti-self-conjugate 
dyadic  vanishes;  but  the  second  does  not.  For  any  such 
dyadic  is  planar  but  cannot  be  linear. 

Nonion  Form.     Determinants}-    Invariants  of  a  Dyadic 
121.]     If  0  be  expressed  in  nonion  form 

<P  =  OII  ii  +  «12ij  +  «i3ik  (13) 

+  a21ji+a22jj  +  «23jk 
+  a31  ki  +  a32  kj  +  a33  k  k. 

The  conjugate  of  0  has  the  same  scalar  coefficients  as  0,  but 
they  are  arranged  symmetrically  with  respect  to  the  main 
diagonal.  Thus 


+  «12  j  i  +  «22  j  j  +  «32  j  k,  (70) 

+  «13ki  +  «23Jk  +  «33kk- 

The  second  of  <P  may  be  computed.  Take,  for  instance,  one 
term.  Let  it  be  required  to  find  the  coefficient  of  ij  in  <P2. 
What  terms  in  0  can  yield  a  double  cross  product  equal  to 
ij?  The  vector  product  of  the  antecedents  must  be  i  and 
the  vector  product  of  the  consequents  must  be  j.  Hence  the 
antecedents  must  be  j  and  k  ;  and  the  consequents,  k  and  i. 
These  terms  are 

x  «33  kk  =  -  aai  a33  i  J 


1  The  results  hold  only  for  determinants  of  the  third  order.    The  extension  to 
determinants  of  higher  orders  is  through  Multiple  Algebra. 


316 


VECTOR  ANALYSIS 


Hence  the  term  in  ij  in  <P2  is 

(a3la23  ~~  a21  a8s)iJ' 

This  is  the  first  minor  of  alz  in  the  determinant 


'11 


"13 


(23 


"88 


This  minor  is  taken  with  the  negative  sign.  That  is,  the 
coefficient  of  i  j  in  <P2  is  what  is  termed  the  cof  actor  of  the 
coefficient  of  ij  in  the  determinant.  The  cof  actor  is  merely 
the  first  minor  taken  with  the  positive  or  negative  sign 
according  as  the  sum  of  the  subscripts  of  the  term  whose 
first  minor  is  under  consideration  is  even  or  odd.  The  co- 
efficient of  any  dyad  in  <P2  is  easily  seen  to  be  the  cofactor  of 
the  corresponding  term  in  0.  The  cofactors  are  denoted 
generally  by  large  letters. 


An  = 


is  the  cofactor  of  a11( 


*M 

*33 


612 
X23 


is  the  cofactor  of  a 


12- 


is  the  cofactor  of  a-,. 


32' 


(71) 


a21         a2 
With  this  notation  the  second  of  <P  becomes 

+  Asl  ki  +  ^32  k  j  +  ^33  kk. 

The  value  of  the  third  of  0  may  be  obtained  by  writing 
as  the  sum  of  three  dyads 

0  =  (ani+a21j  +  a31k)i 


LINEAR    VECTOR  FUNCTIONS 

=  [Oil1  +  «21J  +  *81k)          (a2li  +  «22J 

j  +  a33k)]  [ijk] 


317 


This  is  easily  seen  to  be  equal  to  the  determinant 


'21 


'22 


(32 


33 


(72) 


For  this  reason  03  is  frequently  called  the  determinant  of  0 
and  is  written 

03  =  I  0  I  (72)' 

The  idea  of  the  determinant  is  very  natural  when  0  is 
regarded  as  expressed  in  nonion  form.  On  the  other  hand 
unless  0  be  expressed  in  that  form  the  conception  of  <P3, 
the  third  of  0,  is  more  natural. 

The  reciprocal  of  a  dyadic  in  nonion  form  may  be  found 
most  easily  by  making  use  of  the  identity 

0^0o=^sl  (68) 


or 


or 


Hence   0~l  =  - 


n 


31 


(73) 


318  VECTOR  ANALYSIS 

If  the  determinant  be  denoted  by  D 


If  ¥  is  a  second  dyadic  given  in  nonion  form  as 


+  631ki  +  &32kj  +  633  kk, 

the  product  0  •  ¥  of  the  two  dyadics  may  readily  be  found 
by  actually  performing  the  multiplication 


a!3    32  ll    13         a!2    23         a!3    33 

a21  &11  +  a22  &21  +  a23  J3l)  J  i  +  (a21  &12  +  a2 
+  a23  &32)  J  J  +  (a21  J13  +  a22  J23  +  tt23  &3s 
a31  &11  +  a32  J21  +  «33  &3l)  k  i  +  (a31  &12  +  a3 
+  a33  ^82)  k  J  +  Ca31  &12  +  a32  &28  +  a83  &3s) 


31 


Since  the  third  or  determinant  of  a  product  is  equal  to  the 
product  of  the  determinants,  the  law  of  multiplication  of 
determinants  follows  from  (65)  and  (74). 


LINEAR    VECTOR   FUNCTIONS 


319 


31 


"u 


C21 


[31 


22 


'32 


'23 


•33 


21 


f31 


"12 
^22 


+  a 
+  a 
+  a 


12 


22 


32 


+  #] 

+  a2 

+     «0 


'13 


as 


23 


+  «13    „ 

+  «23  &31 
+  a33  ^31 


(76) 


•M 


32 


The  rule  may  be  stated  in  words.  To  multiply  two  deter- 
minants form  the  determinant  of  which  the  element  in  the 
with  row  and  nth  column  is  the  sum  of  the  products  of  the 
elements  in  the  with  row  of  the  first  determinant  and  nth 
column  of  the  second. 

If  <P  =  al  +  bm  +  cn, 

02  =  bxc  mxn  +  cxa  nxl  +  axb  Ixm. 

=  (02)3=  [b  xc    cxa    axb]     [mxn    nxl     Ixm] 


Then 


Hence 
Hence 


21 


L31 


-12 


L22 


L32 


L13 


L23 


L33 


an     aiz     a 

a21       *22       a< 

a31     a3«     a 


33 


(7T) 


The  determinant  of  the  cofactors  of  a  given  determinant  of 
the  third  order  is  equal  to  the  square  of  the  given  determinant. 
122.]  A  dyadic  0  has  three  scalar  invariants  —  that  is 
three  scalar  quantities  which  are  independent  of  the  form  in 
which  0  is  expressed.  These  are 


the  scalar  of  0,  the  scalar  of  the  second  of  0,  and  the  third 
or  determinant  of  0.  If  0  be  expressed  in  nonion  form  these 
quantities  are 


320 


VECTOR  ANALYSIS 


(78) 


'31 


(22 
''32 


No  matter  in  terms  of  what  right-handed  rectangular  system 
of  these  unit  vectors  0  may  be  expressed  these  quantities  are 
the  same.  The  scalar  of  0  is  the  sum  of  the  three  coefficients 
in  the  main  diagonal.  The  scalar  of  the  second  of  0  is  the 
sum  of  the  first  minors  or  cofactors  of  the  terms  in  the 
main  diagonal  The  third  of  0  is  the  determinant  of  the 
coefficients.  These  three  invariants  are  by  far  the  most 
important  that  a  dyadic  0  possesses. 

Theorem  :  Any  dyadic  satisfies  a  cubic  equation  of  which 
the  three  invariants  0S,  0ZS,  08  are  the  coefficients. 


By  (68) 


-xl\  •  (0  -  xT)c  =  (0  -  * 


(0-xl)s  = 


—  X 


II 


«22  — a;  «2 


—  x 


Hence         (0  —  x  I)3  =  08  —  x  02S  +  x20s  —  x* 
as  may  be  seen  by  actually  performing  the  expansion. 

(0  -  xl\  •  (0  —  x I)c,  =  03  —  x  023  +  xz  0S  —  xs. 

This  equation  is  an  identity  holding  for  all  values  of  the 
scalar  x.  It  therefore  holds,  if  in  place  of  the  scalar  a?,  the 
dyadic  0  which  depends  upon  nine  scalars  be  substituted. 
That  is 

But  the  terms  upon  the  left  are  identically  zero.     Hence 

0*  -  08  0*  +  02S  0  -  0&I  =  0.  (79) 


LINEAR   VECTOR   FUNCTIONS  321 

This  equation  may  be  called  the  Hamilton-Cayley  equation. 
Hamilton  showed  that  a  quaternion  satisfied  an  equation 
analogous  to  this  one  and  Cayley  gave  the  generalization  to 
matrices.  A  matrix  of  the  Tith  order  satisfies  an  algebraic 
equation  of  the  nth  degree.  The  analogy  between  the  theory 
of  dyadics  and  the  theory  of  matrices  is  very  close.  In  fact, 
a  dyadic  may  be  regarded  as  a  matrix  of  the  third  order  and 
conversely  a  matrix  of  the  third  order  may  be  looked  upon  as 
a  dyadic.  The  addition  and  multiplication  of  matrices  and 
dyadics  are  then  performed  according  to  the  same  laws.  A 
generalization  of  the  idea  of  a  dyadic  to  spaces  of  higher 
dimensions  than  the  third  leads  to  Multiple  Algebra  and  the 
theory  of  matrices  of  orders  higher  than  the  third. 

SUMMARY  OF  CHAPTER  V 

A  vector  r'  is  said  to  be  a  linear  function  of  a  vector  r 
when  the  components  of  r'  are  linear  homogeneous  functions 
of  the  components  of  r.  Or  a  function  of  r  is  said  to  be  a 
linear  vector  function  of  r  when  the  function  of  the  sum  of 
two  vectors  is  the  sum  of  the  functions  of  those  vectors. 

f  (ri  +  r2)  =  f  (ri)  +  f  (r2).  (4) 

These  two  ideas  of  a  linear  vector  function  are  equivalent. 
A  sum  of  a  number  of  symbolic  products  of  two  vectors, 
which  are  obtained  by  placing  the  vectors  in  juxtaposition 
without  intervention  of  a  dot  or  cross  and  which  are  called 
dyads,  is  called  a  dyadic  and  is  represented  by  a  Greek 
capital.  A  dyadic  determines  a  linear  vector  function  of 
a  vector  by  direct  multiplication  with  that  vector 

0  =  aj  !>!  +  a2  b2  +  a3  b3  -f  •  •  •  CO 

0  •  r  =  a1b1  •  r  +  a2  b2  •  r  +  a3  b3  •  r  +  •-•      (8) 
21 


322  VECTOR  ANALYSIS 

Two  dyadics  are  equal  when  they  are  equal  as  operators 
upon  all  vectors  or  upon  three  non-coplanar  vectors.  That 
is,  when 

0  •  r  =  W  •  r  for  all  values  or  for  three  non- 

coplanar  values  of  r,  (10) 

or  r  •  0  =  r  •  W  for  all  values  or  for  three  non- 

coplanar  values  of  r, 

or      g»0»r  =  s»?r»r  for  all  values  or  for  three  non- 
coplanar  values  of  r  and  s. 

Any  linear  vector  function  may  be  represented  by  a  dyadic. 

Dyads  obey  the  distributive  law  of  multiplication  with 
regard  to  the  two  vectors  composing  the  dyad 

H  ----  )  =  al  +  am  +  an  +  •  •  • 

+  bl  +  bm  +  bn  +  •  •  • 
•f  cl  +  cm  +  en  +  •  •  • 


(11)' 

Multiplication  by  a  scalar  is  associative.  In  virtue  of  these 
two  laws  a  dyadic  may  be  expanded  into  a  sum  of  nine  terms 
by  means  of  the  fundamental  dyads, 

ii,     ij,     ik, 

ji,    Jj,    jk,  (12) 

ki,    kj,   kk, 
as  0  =  anii  +  a12  ij  +  a13  ik, 

=  «2i  J  i  +  «22  j  j  +  «23  J  k>  (13) 

=  a31ki+  «32kj  +  a33  kk. 

If  two  dyadics  are  equal  the  corresponding  coefficients  in 
their  expansions  into  nonion  form  are  equal  and  conversely. 


LINEAR    VECTOR  FUNCTIONS  323 

Any  dyadic  may  be  expressed  as  the  sum  of  three  dyads  of 
which  the  antecedents  or  the  consequents  are  any  three 
given  non-coplanar  vectors.  This  expression  of  the  dyadic  is 
unique. 

The  symbolic  product  ab  known  as  a  dyad  is  the  most 
general  product  of  two  vectors  in  which  multiplication  by  a 
scalar  is  associative.  It  is  called  the  indeterminate  product. 
The  product  imposes  five  conditions  upon  the  vectors  a  and 
b.  Their  directions  and  the  product  of  their  lengths  are 
determined  by  the  product.  The  scalar  and  vector  products 
are  functions  of  the  indeterminate  product.  A  scalar  and 
a  vector  may  be  obtained  from  any  dyadic  by  inserting  a  dot 
and  a  cross  between  the  vectors  in  each  dyad.  This  scalar 
and  vector  are  functions  of  the  dyadic. 

0a  =  ax  .  bj  +  a2  .  b2  +  a3  .  b3  +  .  .  .        (18) 

<PX  =  aj  x  bx  +  a2  x  b2  +  a3  x  b3  +  .  .  •        (19) 

0j  =  i.0.i  +  j.0.j  +  k.0«k          (20) 

=  a       ~^~  a        '    a' 


+  (i.  0.j-j  .  0.i)k  (21) 

=  <>23  -  «32)  i  +   <>31  -  a13>  J  +   Ol2  -  «2l)  k« 

The  direct  product  of  two  dyads  is  the  dyad  whose  ante- 
cedent and  consequent  are  respectively  the  antecedent  of  the 
first  dyad  and  the  consequent  of  the  second  multiplied  by 
the  scalar  product  of  the  consequent  of  the  first  dyad  and 
the  antecedent  of  the  second. 


(ab)  •  (cd)  =  (b.c)ay.  (23) 

The  direct  product  of  two  dyadics  is  the  formal  expansion, 
according  to  the  distributive  law,  of  the  product  into  the 


324  VECTOR  ANALYSIS 

sum  of  products  of  dyads.  Direct  multiplication  of  dyadics 
or  of  dyadics  and  a  vector  at  either  end  or  at  both  ends  obeys 
the  distributive  and  associative  laws  of  multiplication.  Con- 
sequently such  expressions  as 

0.V.T,     &.<P.¥,     s.tf.r.r,      Q.W.Q      (24X26) 

may  be  written  without  parentheses;  for  parentheses  may 
be  inserted  at  pleasure  without  altering  the  value  of  the 
product.  In  case  the  vector  occurs  at  other  positions  than 
at  the  end  the  product  is  no  longer  associative. 

The  skew  product  of  a  dyad  and  a  vector  may  be  defined 

by  the  equation 

(ab)  x  r  =  a  b  x  r, 

r  x  (ab)  =  r  x  a  b.  (28) 

The  skew  product  of  a  dyadic  and  a  vector  is  equal  to  the 
formal  expansion  of  that  product  into  a  sum  of  products  of 
dyads  and  that  vector.  The  statement  made  concerning  the 
associative  law  for  direct  products  holds  when  the  vector  is 
connected  with  the  dyadics  in  skew  multiplication.  The 
expressions 

r  x  <P  •  ?F,      4>.¥xr,     r  x  <P  •  s,     r  •  $  x  s,     r  x  0  x  s   (29) 

may  be  written  without  parentheses  and  parentheses  may  be 
inserted  at  pleasure  without  altering  the  value  of  the  product. 
Moreover 

s  .  (r  x  <P)  =  (s  x  r)  •  0,     (  d>  x  r)  •  s  =  0  •  (r  x  s), 

0 .  (r  x  5T)  =  (<P  x  r)  .  ¥.  (31)' 

But  the  parentheses  cannot  be  omitted. 

The  necessary  and  sufficient  condition  that  a  dyadic  may 
be  reduced  to  the  sum  of  two  dyads  or  to  a  single  dyad  or 
to  zero  is  that,  when  expressed  as  the  sum  of  three 
dyads  of  which  the  antecedents  (or  consequents)  are  known 


LINEAR    VECTOR  FUNCTIONS  325 

to  be  non-coplanar,  the  consequents  (or  antecedents)  shall 
be  respectively  coplanar  or  collinear  or  zero.  A  complete 
dyadic  is  one  which  cannot  be  reduced  to  a  sum  of  fewer 
than  three  dyads.  A  planar  dyadic  is  one  which  can  be 
reduced  to  a  sum  of  just  two  dyads.  A  linear  dyadic  is  one 
which  can  be  reduced  to  a  single  dyad. 

A  complete  dyadic  possesses  no  degree  of  nullity.  There 
is  no  direction  in  space  for  which  it  is  an  annihilator.  A 
planar  dyadic  possesses  one  degree  of  nullity.  There  is  one 
direction  in  space  for  which  it  is  an  annihilator  when  used  as 
a  prefactor  and  one  when  used  as  a  postfactor.  A  linear 
dyadic  possesses  two  degrees  of  nullity.  There  are  two 
independent  directions  in  space  for  which  it  is  an  annihilator 
when  used  as  a  prefactor  and  two  directions  when  used  as  a 
postfactor.  A  zero  dyadic  possesses  three  degrees  of  nullity 
or  complete  nullity.  It  annihilates  every  vector  in  space. 

The  products  of  a  complete  dyadic  and  a  complete,  planar, 
or  linear  dyadic  are  respectively  complete,  planar,  or  linear. 
The  products  of  a  planar  dyadic  with  a  planar  or  linear  dyadic 
are  respectively  planar  or  linear,  except  in  certain  cases  where 
relations  of  perpendicularity  between  the  consequents  of  the 
first  dyadic  and  the  antecedents  of  the  second  introduce  one 
more  degree  of  nullity  into  the  product.  The  product  of  a 
linear  dyadic  by  a  linear  dyadic  is  in  general  linear ;  but  in 
case  the  consequent  of  the  first  is  perpendicular  to  the  ante- 
cedent of  the  second  the  product  vanishes.  The  product  of 
any  dyadic  by  a  zero  dyadic  is  zero. 

A  dyadic  which  when  applied  to  any  vector  in  space  re- 
produces that  vector  is  called  an  idemfactor.  All  idemfactors 
are  equal  and  reducible  to  the  form 

I  =  ii  +  jj  +  kk.  (33) 

Or  I  =  aa'  +  bb'  +  cc'.  (34) 

The  product  of  any  dyadic  and  an  idemfactor  is  that  dyadic. 


326  VECTOR   ANALYSIS 

If  the  product  of  two  complete  dyadics  is  equal  to  the  idem- 
factor  the  dyadics  are  commutative  and  either  is  called 
the  reciprocal  of  the  other.  A  complete  dyadic  may  be 
canceled  from  either  end  of  a  product  of  dyadics  and  vectors 
as  in  ordinary  algebra  ;  for  the  cancelation  is  equivalent  to 
multiplication  by  the  reciprocal  of  that  dyadic.  Incomplete 
dyadics  possess  no  reciprocals.  They  correspond  to  zero  in 
ordinary  algebra.  The  reciprocal  of  a  product  is  equal  to  the 
product  of  the  reciprocals  taken  in  inverse  order. 

(0.  gr)-i  =  sr-i.  0-i.  (38) 

The  conjugate  of  a  dyadic  is  the  dyadic  obtained  by  inter- 
changing the  order  of  the  antecedents  and  consequents.  The 
conjugate  of  a  product  is  equal  to  the  product  of  the  con- 
jugates taken  in  the  opposite  order. 

(0.  ¥)„=  ¥c.  4>c.  (40) 

The  conjugate  of  the  reciprocal  is  equal  to  the  reciprocal  of 
the  conjugate.  A  dyadic  may  be  divided  in  one  and  only 
one  way  into  the  sum  of  two  parts  of  which  one  is  self- 
conjugate  and  the  other  anti-self-conjugate. 

0  =  ^(0+  0c)  +  2L(0-0c)-  (43) 

• 

Any  anti-self-conjugate  dyadic  or  the  anti-self-conjugate 
part  of  any  dyadic,  used  in  direct  multiplication,  is  equivalent 
to  minus  one-half  the  vector  of  that  dyadic  used  in  skew 
multiplication. 


]r.(0-0ff)=-lrx0x.  (44) 

A  dyadic  of  the  form  c  X  I  or  I  x  c  is  anti-self-conjugate  and 
used  in  direct  multiplication  is  equivalent  to  the  vector  c 
used  in  skew  multiplication. 


LINEAR    VECTOR  FUNCTIONS  327 

Also  o  x  r  =  (I  x  o)  •  r  ==  (o  x  I)  •  r,  (46) 

c  x  4>  =  (I  x  c)  •  0  =  (c  x  I)  •  0. 

The  dyadic  c  X  I  or  I  x  c,  where  c  is  a  unit  vector  is  a  quad- 
rantal  versor  for  vectors  perpendicular  to  c  and  an  annihilator 
for  vectors  parallel  to  c.  The  dyadic  Ixc  +  cc  is  a  true 
quadrantal  versor  for  all  vectors.  The  powers  of  these  dyadics 
behave  like  the  powers  of  the  imaginary  unit  V  —  1,  as  may 
be  seen  from  the  geometric  interpretation.  Applied  to  the 
unit  vectors  i,  j,  k 

I  x  i  =  i  x  I  =  kj  —  j  k,  etc.  -(49) 

The  vector  a  x  b  in  skew  multiplication  is  equivalent  to 
(a  X  b)  x  I  in  direct  multiplication. 

(a  x  b)  x  I  =  1  x  (a  x  b)  =  b  a  —  ab        (50) 
(a  x  b)  x  r  =  (b  a  —  a  b)  •  r 
r  x  (ax  b)=r  •  (ba-ab).  (51) 

A  complete  dyadic  may  be  reduced  to  a  sum  of  three 
dyads  of  which  the  antecedents  among  themselves  and  the 
consequents  among  themselves  each  form  a  right-handed 
rectangular  system  of  three  unit  vectors  and  of  which  the 
scalar  coefficients  are  all  positive  or  all  negative. 

0=±  (ai'i  +  &j'j  +  ck'k).  (53) 

This  is  called  the  normal  form  of  the  dyadic.  An  incom- 
plete dyadic  may  be  reduced  to  this  form  but  one  or  more  of 
the  coefficients  are  zero.  The  reduction  is  unique  in  case 
the  constants  a,  6,  c  are  different.  In  case  they  are  not 
different  the  reduction  may  be  accomplished  in  more  than 
one  way.  Any  self-conjugate  dyadic  may  be  reduced  to 

the  normal  form 

&  =  aii  +  6jj  +  ckk,  (55) 

in  which  the  constants  a,  &,  c  are  not  necessarily  positive. 


328  VECTOR  ANALYSIS 

The  double  dot  and  double  cross  multiplication  of  dyads 
is  defined  by  the  equations 

a  b  :  c  d  =  a  •  c    b.d,  (56) 

ab£cd  =  axc    bxd.  (57) 

The  double  dot  and  double  cross  multiplication  of  dyadics 
is  obtained  by  expanding  the  product  formally,  according  to 
the  distributive  law,  into  a  sum  of  products  of  dyads.  The 
double  dot  and  double  cross  multiplication  of  dyadics  is  com- 
mutative but  not  associative. 

One-half  the  double  cross  product  of  a  dyadic  0  by  itself 
is  called  the  second  of  0.  If 

<P  =  al  +  bm  +  cn, 
02  =  10x0  =  bxc  mxn+cxa  nxl  +  axb  Ixm.   (61) 

One-third  of  the  double  dot  product  of  the  second  of  0  and  0 
is  called  the  third  of  0  and  is  equal  to  the  product  of  the 
scalar  triple  product  of  the  antecedents  of  0  and  the  scalar 
triple  product  of  the  consequent  of  0. 

0Z  =  \0*  0:  0=[abc]  [1m  n].  (62) 

The  second  of  the  conjugate  is  the  conjugate  of  the  second. 
The  third  of  the  conjugate  is  equal  to  the  third  of  the 
original  dyadic.  The  second  and  third  of  the  reciprocal  are 
the  reciprocals  of  the  second  and  third  of  the  second  and 
third  of  a  dyadic.  The  second  and  third  of  a  product  are  the 
products  of  the  seconds  and  thirds. 


(0.3r),=  0a.JTa  (65) 

(*•  »).!->>, 


LINEAR    VECTOR  FUNCTIONS  329 

The  product  of  the  second  and  conjugate  of  a  dyadic  is  equal 
to  the  product  of  the  third  and  the  idemfactor. 

0^'0C=03I  (68) 

The  conditions  for  the  various  degrees  of  nullity  may  be 
expressed  in  terms  of  the  second  and  third  of  0. 

0g  •£  0,     0  is  complete 
03  =  0,  02  *  0,     0  is  planar  (69) 

03  =  0,  02  =  0,  0  #  0,     0  is  linear. 

The  closing  sections  of  the  chapter  contain  the  expressions 
(70)-(78)  of  a  number  of  the  results  in  nonion  form  and  the 
deduction  therefrom  of  a  number  of  theorems  concerning 
determinants.  They  also  contain  the  cubic  equation  which  is 
satisfied  by  a  dyadic  0. 

0s  —  0S  02  +  0  s  03  +  0q  1  =  0.          (79) 

This  is  called  the  Hamilton-Cayley  equation.  The  coeffi- 
cients 0S,  <P2/S,  and  03  are  the  three  fundamental  scalar  in- 
variants of  0. 

EXERCISES  ON  CHAPTER  V 

1.  Show  that  the  two   definitions   given  in  Art.   98  for 
a  linear  vector  function  are  equivalent 

2.  Show  that  the  reduction  of  a  dyadic  as  in  (15)  can  be 
accomplished  in  only  one  way  if  a,  b,  c,  1,  m,  n,  are  given. 

3.  Show  (0  X  &)c=  —  a  x  && 

4.  Show  that  if  0xr  =  ¥xr  for  any  value  of  r  different 
from  zero,  then  0  must  equal   SF— unless  both  0  and  W  are 
linear  and  the  line  of  their  consequents  is  parallel  to  r. 

5.  Show  that  if  0  •  r  =  0  for  any  three  non-coplanar  values 
of  r,  then  0  =  0. 


330  VECTOR  ANALYSIS 

6.  Prove  the  statements  made  in  Art.  106  and  the  con- 
verse of  the  statements. 

7.  Show  that  if  Q  is  complete  and  if  <P  •  Q  =  ¥  •  Q ,  then 
<P  and   ¥  are  equal.     Give  the  proof  by  means  of   theory 
developed  prior  to  Art.  109. 

8.  Definition :  Two  dyadics  such  that  &•¥=  ¥•  0  —  that 
is  to  say,  two  dyadics  that  are  commutative  —  are  said  to  be 
homologous.    Show  that  if  any  number  of  dyadics  are  homoge- 
neous to  one  another,  any  other  dyadics  which  may  be  obtained 
from  them  by  addition,  subtraction,  and  direct  multiplication 
are  homologous  to  each  other  and  to  the  given  dyadics.    Show 
also  that  the  reciprocals  of  homologous  dyadics  are  homolo- 
gous.    Justify  the  statement  that  if    0  •  ¥~l   or    ¥~l  •  0, 
which  are  equal,  be  called  the  quotient  of  0  by  3F,  then  the 
rules    governing    addition,   subtraction,   multiplication    and 
division  of  homologous  dyadics  are  identical  with  the  rules 
governing  these   operations  in  ordinary  algebra  —  it  being 
understood  that  incomplete  dyadics  are  analogous  to  zero, 
and  the  idemfactor,  to  unity.     Hence  the  algebra  and  higher 
analysis  of  homologous  dyadics  is  practically  identical  with 
that  of  scalar  quantities. 

9.     Show  that  (I  x  c)  •  0  =  c  x  ¥  and  (c  x  I)  •  <P  =  c  x  0. 

10.  Show  that  whether  or  not  a,  b,  c  be  coplanar 

abxc+bcxa+c  a  x  b  =  [a  b  c]  I 
and  bxca+cxab+axbc=[abc]I. 

11.  If  a,  b,  c  are  coplanar  use  the  above  relation  to  prove 
the  law  of  sines  for  the  triangle  and  to  obtain  the  relation 
with  scalar  coefficients  which  exists  between  three  coplanar 
vectors.     This  may  be  done  by  multiplying  the  equation  by  a 
unit  normal  to  the  plane  of  a,  b,  and  c. 

12.  What  is  the  condition  which  must  subsist  between  the 
coefficients  in  the  expansion  of  a  dyadic  into  nonion  form  if 


LINEAR    VECTOR  FUNCTIONS  331 

the  dyadic  be  self -conjugate?     What,  if  the  dyadic  be  anti- 
self -conjugate  ? 

13.  Prove  the  statements  made  in  Art.  116  concerning  the 
number  of  ways  in  which  a  dyadic  may  be   reduced  to  its 
normal  form. 

14.  The  necessary   and  sufficient  condition  that  an  anti- 
self-conjugate  dyadic   0  be  zero  is  that  the  vector  of  the 
dyadic  shall  be  zero. 

15.  Show  that  if  0  be  any  dyadic  the  product  0  •  <PC  is 
self-conjugate. 

16.  Show  how  to  make  use  of   the  relation   <PX  =  0  to 
demonstrate  that  the  antecedents  and  consequents  of  a  self- 
conjugate  dyadic  are  the  same  (Art.  116). 

17.  Show  that  4>z  $  0Z  =  4>*  <P 

and  (<P  +  W\  =  <P2  +  0  £  V  +  Wy 

18.  Show  that  if  the  double  dot  product  <P  :  0  of  a  dyadic 
by  itself  vanishes,  the  dyadic  vanishes.     Hence   obtain  the 
condition  for  a  linear  dyadic  in  the  form  <PZ  :  <PZ  =  0. 

19.  Show  that     (0  +  e f)3  =  4>3  +  e •  <P2  •  f. 

20.  Show  that  (0  +  ¥)3  =  03  +  02  :  V  +  0  :  Vz  +  ¥s. 

21.  Show  that  the  scalar  of  a  product  of  dyadics  is  un- 
changed by  cyclic  permutation  of  the  dyadics.     That  is 


CHAPTER  VI 

ROTATIONS  AND  STRAINS 

123.]  IN  the  foregoing  chapter  the  analytical  theory  of 
dyadics  has  been  dealt  with  and  brought  to  a  state  of 
completeness  which  is  nearly  final  for  practical  purposes. 
There  are,  however,  a  number  of  new  questions  which  present 
themselves  and  some  old  questions  which  present  themselves 
under  a  new  form  when  the  dyadic  is  applied  to  physics 
or  geometry.  Moreover  it  was  for  the  sake  of  the  applica- 
tions of  dyadics  that  the  theory  of  them  was  developed.  It  is 
then  the  object  of  the  present  chapter  to  supply  an  extended 
application  of  dyadics  to  the  theory  of  rotations  and  strains 
and  to  develop,  as  far  as  may  appear  necessary,  the  further 
analytical  theory  of  dyadics. 

That  the  dyadic  $  may  be  used  to  denote  a  transformation 
of  space  has  already  been  mentioned.  A  knowledge  of  the 
precise  nature  of  this  transformation,  however,  was  not  needed 
at  the  time.  Consider  r  as  drawn  from  a  fixed  origin,  and  r' 
as  drawn  from  the  same  origin.  Let  now 

r'  =  0-r. 

This  equation  therefore  may  be  regarded  as  defining  a  trans- 
formation of  the  points  P  of  space  situated  at  the  terminus  of 
r  into  the  point  P',  situated  at  the  terminus  of  r'.  The  origin 
remains  fixed.  Points  in  the  finite  regions  of  space  remain  in 
the  finite  regions  of  space.  Any  point  upon  a  line 

r  =  b  4-  x  a 
becomes  a  point  r'  =  0  •  b  -f  x  0  •  a. 


ROTATIONS  AND  STRAINS  333 

Hence  straight  lines  go  over  into  straight  lines  and  lines 
parallel  to  the  same  line  a  go  over  by  the  transformation  into 
lines  parallel  to  the  same  line  0  •  a.  In  like  manner  planes 
go  over  into  planes  and  the  quality  of  parallelism  is  invariant. 
Such  a  transformation  is  known  as  a  homogeneous  strain. 
Homogeneous  strain  is  of  frequent  occurrence  in  physics.  For 
instance,  the  deformation  of  the  infinitesimal  sphere  in  a  fluid 
(Art.  76)  is  a  homogeneous  strain.  In  geometry  the  homo- 
geneous strain  is  generally  known  by  different  names.  It  is 
called  an  affine  collineation  with  the  origin  fixed.  Or  it  is 
known  as  a  linear  homogeneous  transformation.  The  equa- 
tions of  such  a  transformation  are 


=  a 


n 


y'  = 


z'  =  asix  +  a^y  +  «33«. 

124.]  Theorem  :  If  the  dyadic  0  gives  the  transformation 
of  the  points  of  space  which  is  due  to  a  homogeneous  strain, 
<P2,  the  second  of  <P,  gives  the  transformation  of  plane  areas 
which  is  due  to  that  strain  and  all  volumes  are  magnified  by 
that  strain  in  the  ratio  of  $3,  the  third  or  determinant  of  0 
to  unity. 

Let  <p  =  al  +  bm  +  en 


The  vectors  1',  m',  n'  are  changed  by  0  into  a,  b,  c.  Hence 
the  planes  determined  by  m'  and  n',  n'  and  1',  1'  and  m'  are 
transformed  into  the  plane  determined  by  b  and  c,  c  and  a, 
a  and  b.  The  dyadic  which  accomplishes  this  result  is 

(ft,  =  b  x  c  mxn  +  cxa  nxl  +  axb  Ixm. 

i         A 

Hence  if  s  denote  any  plane  area  in  space,  the  transformation 
due  to  0  replaces  s  by  the  area  s'  such  that 

s'  =  0  •  s. 


334  VECTOR  ANALYSIS 

It  is  important  to  notice  that  the  vector  s  denoting  a  plane 
area  is  not  transformed  into  the  same  vector  s'  as  it  would 
be  if  it  denoted  a  line.  This  is  evident  from  the  fact  that  in 
the  latter  case  0  acts  on  s  whereas  in  the  former  case  02  acts 
upon  s. 

To  show  that  volumes  are  magnified  in  the  ratio  of  <#3  to 
unity  choose  any  three  vectors  d,  e,  f  which  determine  the 
volume  of  a  parallelepiped  [d  e  f].  Express  0  with  the  vec- 
tors which  form  the  reciprocal  system  to  d,  e,  f  as  consequents. 


The  dyadic  0  changes  d,  e,  f  into  a,  b,  c  (which  are  different 
from  the  a,  b,  c  above  unless  d,  e,  f  are  equal  to  1',  m',  n'). 
Hence  the  volume  [d  e  f  ]  is  changed  into  the  volume  [a  b  c]. 

[d'e'f] 


Hence  [a  be]  =  [def]  3>y 

The  ratio  of  the  volume  [a  b  cj  to  [d  e  f]  is  as  03  is  to  unity. 
But  the  vectors  d,  e,  f  were  any  three  vectors  which  deter- 
mine a  parallelepiped.  Hence  all  volumes  are  changed  by 
the  action  of  0  in  the  same  ratio  and  this  ratio  is  as  $3  is  to  1. 

dotations  about  a  Fixed  Point.     Versors 

125.]  Theorem  :  The  necessary  and  sufficient  condition  that 
a  dyadic  represent  a  rotation  about  some  axis  is  that  it  be 
reducible  to  the  form 

0  =  i'i+j'j  +  k'k  (1) 

where  i',  j',  k'  and  i,  j,  k  are  two  right-handed  rectangular 
systems  of  unit  vectors. 

Let  r  =  a>i  +       +  2k 


ROTATIONS  AND  STRAINS  335 

Hence  if  0  is  reducible  to  the  given  form  the  vectors  i,  j,  k 
are  changed  into  the  vectors  i',  j',  k'  and  any  vector  r  is 
changed  from  its  position  relative  to  i,  j,  k  into  the  same  posi- 
tion relative  to  i',j',k'.  Hence  by  the  transformation  no 
change  of  shape  is  effected.  The  strain  reduces  to  a  rotation 
which  carries  i,  j,  k  into  i',j',  k'.  Conversely  suppose  the 
body  suffers  no  change  of  shape  —  that  is,  suppose  it  subjected 
to  a  rotation.  The  vectors  i,  j,  k  must  be  carried  into  another 
right-handed  rectangular  system  of  unit  vectors.  Let  these 
be  i',  j',  k'.  The  dyadic  0  may  therefore  be  reduced  to  the 

form 

0  =  i'  i  +  j'  j  +  k'  k. 

Definition  :  A  dyadic  which  is  reducible  to  the  form 
i'i  +  j'j  +  k'k 

and  which  consequently  represents  a  rotation  is  called  a 
versor. 

Theorem:  The  conjugate  and  reciprocal  of  a  versor  are 
equal,  and  conversely  if  the  conjugate  and  reciprocal  of  a 
dyadic  are  equal  the  dyadic  reduces  to  a  versor  or  a  versor 
multiplied  by  the  negative  sign. 

Let  <P  =  i'i+j'j  +  k'k, 

0c  =  ii'+jj'  +  kk', 
0.  0c  =  i'i'  +  j'j'  +  k'k'  =  I 
0-*=  0c. 

Hence  the  first  part  of  the  theorem  is  proved.     To  prove  the 

second  part  let 

0  =  ai  +  b  j  +  ck, 

<Pc  =  ia  +  j  b  +  kc, 


If  fl-^00        0-0c 

Hence  aa  +  bb  +  cc  =  I. 


336  VECTOR  ANALYSIS 

Hence  (Art.  108)  the  antecedents  a,  b,  c  and  the  consequents 
a,  b,  c  must  be  reciprocal  systems.  Hence  (page  87)  they 
must  be  either  a  right-handed  or  a  left-handed  rectangular 
system  of  unit  vectors.  The  left-handed  system  may  be 
changed  to  a  right-handed  one  by  prefixing  the  negative 
sign  to  each  vector.  Then 

0  =  i'i  +  j'  j  +  k'k, 
or  0  =  -(i'i  +  j' j  +  k'k). 

The  third  or  determinant  of  a  versor  is  evidently  equal  to 
unity;  that  of  the  versor  with  a  negative  sign,  to  minus  one. 
Hence  the  criterion  for  a  versor  may  be  stated  in  the  form 

0.0C=  I,          08=I0I=1.  (2) 

Or  inasmuch  as  the  determinant  of  0  is  plus  or  minus  one 
if  0.  0C=I,  it  is  only  necessary  to  state  that  if 

0.0C=I,       03=I0I>0,  (2)' 

0  is  a  versor. 

There  are  two  geometric  interpretations  of  the  transforma- 
tion due  to  a  dyadic  0  such  that 

0  •  0C  =  I       03  =  I  0  I  =  -  1  (3) 

0  =  -(i'i  +  j'  j  +  k'k). 

The  transformation  due  to  0  is  one  of  rotation  combined  with 
reflection  in  the  origin.  The  dyadic  i'i+j'j  +  k'k  causes  a 
rotation  about  a  definite  axis  —  it  is  a  versor.  The  negative 
sign  then  reverses  the  direction  of  every  vector  in  space  and 
replaces  each  figure  by  a  figure  symmetrical  to  it  with  respect 
to  the  origin.  By  reversing  the  directions  of  i'  and  j'  the 
system  i',  j',  k'  still  remains  right-handed  and  rectangular, 
but  the  dyadic  takes  the  form 

0  =  i'i+j'j  -k'k, 
or  0  =  (i'i'  +  j'j'-k'k')  •  (i'i+j' j  +  k'k). 


ROTATIONS  AND  STRAINS  337 

Hence  the  transformation  due  to  0  is  a  rotation  due  to 
i'i+j '  j  +  k'k  followed  by  a  reflection  in  the  plane  of  i'  and 
j'.  For  the  dyadic  i'i'  +  j'j'  —  k'k'  causes  such  a  transfor- 
mation of  space  that  each  point  goes  over  into  a  point  sym- 
metrically situated  to  it  with  respect  to  the  plane  of  i'  and  j'. 
Each  figure  is  therefore  replaced  by  a  symmetrical  figure. 

Definition :  A  transformation  that  replaces  each  figure  by 
a  symmetrical  figure  is  called  a  perversion  and  the  dyadic 
which  gives  the  transformation  is  called  a  perversor. 

The  criterion  for  a  perversor  is  that  the  conjugate  of  a 
dyadic  shall  be  equal  to  its  reciprocal  and  that  the  determi- 
nant of  the  dyadic  shall  be  equal  to  minus  one. 

0.0C=I,          I0I=-1.  (3) 

Or  inasmuch  as  if  0  •  <PC  =  I,  the  determinant  must  be  plus 
or  minus  one  the  criterion  may  take  the  form 

0  .  0C  =  I,         I  0  \<  0,  (3)' 

0  is  a  perversor. 

It  is  evident  from  geometrical  considerations  that  the  prod- 
uct of  two  versors  is  a  versor ;  of  two  perversors,  a  versor ; 
but  of  a  versor  and  a  perversor  taken  in  either  order,  a 
perversor. 

126.]  If  the  axis  of  rotation  be  the  i-axis  and  if  the  angle 
of  rotation  be  the  angle  q  measured  positive  in  the  positive 
trigonometric  direction,  then  by  the  rotation  the  vectors 
i,j,  k  are  changed  into  the  vectors  i',j',  k'  such  that 

i'  =  i' 

j'  —  j  cos  q  +  k  sin  q, 
k'  =  —  j  sin  q  +  k  cos  q. 

The  dyadic  0  =  i'i+j'j  +  k'k  which  accomplishes  this  rota- 
tion is 

22 


338  VECTOR  ANALYSIS 

<P  =  i  i  +  cos  q  (j  j  +  k  k)  +  sin  q  (k  j  —  j  k).      (4) 
jj  +  kk  =  I-ii, 
kj  —  jk  =  I  x  i. 
Hence         0  =  i  i  +  cos  q  (I  —  i  i)  +  sin  q  I  x  i.  (5) 

If  more  generally  in  place  of  the  i-axis  any  axis  denoted 
by  the  unit  vector  a  be  taken  as  the  axis  of  rotation  and  if  as 
before  the  angle  of  rotation  about  that  axis  be  denoted  by  <?, 
the  dyadic  0  which  accomplishes  the  rotation  is 

<P  =  a  a  +  cos  q  (I  —  a  a)  +  sin  q  I  x  a.        (6) 

To  show  that  this  dyadic  actually  does  accomplish  the 
rotation  apply  it  to  a  vector  r.  The  dyad  a  a  is  an  idemfactor 
for  all  vectors  parallel  to  a;  but  an  annihilator  for  vectors 
perpendicular  to  a.  The  dyadic  I  —  a  a  is  an  idemfactor 
for  all  vectors  in  the  plane  perpendicular  to  a;  but  an 
annihilator  for  all  vectors  parallel  to  a.  The  dyadic  I  x  a 
is  a  quadrantal  versor  (Art.  113)  for  vectors  perpendicular 
to  a;  but  an  annihilator  for  vectors  parallel  to  a.  If  then 

r  be  parallel  to  a 

<^«r  =  aa«r  =  r. 

Hence  0  leaves  unchanged   all  vectors  (or  components   of 
vectors)  which  are  parallel  to  a.     If  r  is  perpendicular  to  a 

d>  •  r  =  cos  q  r  +  sin  q  a  x  r. 

Hence  the  vector  r  has  been  rotated  in  its  plane  through  the 
angle  q.  If  r  were  any  vector  in  space  its  component  parallel 
to  a  suffers  no  change ;  but  its  component  perpendicular  to  a 
is  rotated  about  a  through  an  angle  of  q  degrees.  The  whole 
vector  is  therefore  rotated  about  a  through  that  angle. 
Let  a  be  given  in  terms  of  i,  j,  k  as 

a  =  ax  i  +  a2  j  +  «3k, 
aa=  a2  ii  +  aa    ij  +  0    «    ik 


ROTATIONS  AND  STRAINS  339 

+  «2al  J*  +  «22  JJ+  «2a3  J* 
+  azal  ki  +  a3a2  kj  +  «32  kk, 

I  =  ii+jj  +  kk, 
I  x  a  =  0ii  —  «3ij  +  «2ik, 

+  «3Ji  +  °JJ-«iJk, 

-«2ki  +  ^kj  +  Okk. 
Hence 

0  =  {a^  (1  —  cos  j)  +  cos  £>  i  i 

+  {a1«2  (1  — cos  q)  —  «3  sin  g-}  i  j 
+  {a1  a3  (1  —  cos  #)  +  «2  sin  3.1  *k 
+  {«2ai  (1  —  cos 2)  +  a3sing-}  ji 
+  (a22  (1  —  cos  q)  +  cos  q}  j  j 

+  {#2  as  (1  ~  cos  #)  ~~  ai  sin  ?}  j  k 

+  {agffj  (1  —  cos  ^)  —  a2  sin  q}  ki 

+  (as  a2  (1  —  cos  ^)  +  ax  sin  q}  kj 

+  (a32  (1  -f-  cos  q)  +  cos  q}  k  k.  (7) 

127.]     If  0  be  written  as  in  equation  (4)  the  vector  of  0 
and  the  scalar  of  0  may  be  found. 

0X  =  i  x  i  +  cos  q  Q  x  j  +  k  x  k)  -f  sin  q  (k  x  j  —  j  x  k) 

0X  =  —  2  sin  q  i 

Qs  =  i  •  i  +  cos  q  (j  •  j  +  kk)  +  sin  q  (k  •  j  —  j  •  k), 
0a  =  1  +  2  cos  q. 

The  axis  of  rotation  i  is  seen  to  have  the  direction  of  —  <PX, 
the  negative  of  the  vector  of  0.  This  is  true  in  general. 
The  direction  of  the  axis  of  rotation  of  any  versor  is  the 
negative  of  the  vector  of  0.  The  proof  of  this  statement 
depends  on  the  invariant  property  of  <#x.  Any  versor  0 
may  be  reduced  to  the  form  (4)  by  taking  the  direction  of  i 


340  VECTOR  ANALYSIS 

coincident  with  the  direction  of  the  axis  of  rotation.  After 
this  reduction  has  been  made  the  direction  of  the  axis  is  seen 
to  be  the  negative  of  <PX.  But  <Py  is  not  altered  by  the 
reduction  of  0  to  any  particular  form  —  nor  is  the  axis  of 
rotation  altered  by  such  a  reduction.  Hence  the  direction  of 
the  axis  of  rotation  is  always  coincident  with  —  <PX,  the  direc- 
tion of  the  negative  of  the  vector  of  <#. 

The  tangent  of  one-half  the  angle  of  version  q  is 


sm  q 


..        »  • 

1  +  cos  q         1  +  0s 

The  tangent  of  one-half  the  angle  of  version  is  therefore 
determined  when  the  values  of  0X  and  @s  are  known.  The 
vector  0X  and  the  scalar  4>s,  which  are  invariants  of  $,  deter- 
mine completely  the  versor  0.  Let  ft  be  a  vector  drawn 
in  the  direction  of  the  axis  of  rotation.  Let  the  magnitude 
of  Q,  be  equal  to  the  tangent  of  one-half  the  angle  q  of 
version. 


The  vector  ft  determines  the  versor  0  completely.     ft  will  be 
called  the  vector  semi-tangent  of  version. 

By  (6)  a  versor  0  was  expressed  in  terms  of  a  unit  vector 
parallel  to  the  axis  of  rotation. 

0  =  a  a  +  cos  q  (I  —  a  a)  +  sin  q  I  x  a. 
Hence  if  ft  be  the  vector  semi-tangent  of  version 


cos      1  -  sin    '  x 


There  is  a  more  compact  expression  for  a  versor  0  in  terms 
of  the  vector  semi-tangent  of  version.  Let  c  be  any  vector  in 
space.  The  version  represented  by  ft  carries 

c  —  ft  x  c  into  c  +  ft  x  c. 


ROTATIONS  AND  STRAINS  341 

It  will  be  sufficient  to  show  this  in  case  c  is  perpendicular  to 
ft.  For  if  c  (or  any  component  of  it)  were  parallel  to  ft  the 
result  of  multiplying  by  ft  x  would  be  zero  and  the  statement 
would  be  that  c  is  carried  into  c.  In  the  first  place  the  mag- 
nitudes of  the  two  vectors  are  equal.  For 

(c  —  ft  x  c)  •  (c  —  ftxc)—  c»c  +  ftxc'ftxc  —  2  c  •  ft  x  c 


C'C  +  ftXC«ftXC  =  C«C  +  ft.ft      C»C  — ft«C      ft«C. 

Since  ft  and  c  are  by  hypothesis  perpendicular 

c«c  +  ftxc.ftxc  =  c2  (1+  tan2  i  q) . 

The  term  c  •  ft  X  c  vanishes.      Hence  the  equality.      In  the 
second  place  the  angle  between  the  two  vectors  is  equal  to  q. 

(c  —  ftxc).(c  +  ftxc)_c«c  —  ftxc»ftxc 
c2  (1  +  tan2  \  q}  c2  (1  +  tan2  \  q) 

c2  (1  —  tan2  -  q) 

'- =  COS  q 

c2  (1  +  tan2  i  q) 

40 

(c  —  ft  x  c)  x  (c  +  ft  x  c)  _       2  c  x  (ft  x  c) 
c2  (1  +  tan2  1  q)  c2  (1  +  tan2  I  q) 

2  c2  tan  1  q 

=  sin  q. 


Hence  the  cosine  and  sine  of  the  angle  between  c  —  ft  x  c 
and  c  +  ft  x  c  are  equal  respectively  to  the  cosine  and  sine  of 
the  angle  q :  and  consequently  the  angle  between  the  vectors 
must  equal  the  angle  q.  Now 


342  VECTOR  ANALYSIS 

c  —  ft  x  c  =  (I  —  I  x  ft)  •  c 
and  (c  +  Q  x  c)  =  (I  +  I  x  ft)  •  c 

(i  +  1  x  a),  (i  -  1  x  a)-1-  (i  -ixft)=i+ixft. 

Multiply  by  c 

(i  +  ixft)«(i-ix  a)-1  •  (c  -  a  x  c)  =  c  +  ft  x  c. 

Hence  the  dyadic 

<P  =  (I  +  I  x  ft)  .  (I  -  I  x  ft)-1  (10)' 

carries  the  vector  c  —  ft  x  c  into  the  vector  c  +  ft  x  c  no  matter 
what  the  value  of  c.  Hence  the  dyadic  <P  determines  the 
version  due  to  the  vector  semi-tangent  of  version  ft. 

The    dyadic   I  +  1  x  ft   carries   the   vector  c  —  ft  x  c    into 
(I  +  ft-ft)c. 

(I  +  Ixft)«(c  —  ftxc)  =  c  +  ftxc  —  ftxc  —  ftx(ftxc) 

(I  +  I  x  ft).(c-ft  +  c)=c  +  ft-ftc  =  (l  +  ft«ft)c. 
Hence  the  dyadic 


carries  the  vector  c  —  ft  x  c  into  the  vector  c,  if  c  be  perpen- 
dicular to  ft  as  has  been  supposed.  Consequently  the  dyadic 

(I  +  Ixft)2 
1  +  ft-ft 

produces  a  rotation  of  all  vectors  in  the  plane  perpendicular 
to  ft.  If,  however,  it  be  applied  to  a  vector  x  ft  parallel  to  ft 
the  result  is  not  equal  to  #  ft. 

(I  +  I  X  ft)  •  (I  +  I  x  ft)  (I  +  I  x  ft)  n          xQ, 

xQ,  =  x  -  -  -ft  = 


1+ft-ft  1+ft-ft  1+ft.ft 


ROTATIONS  AND  STRAINS  343 

To  obviate  this  difficulty  the  dyad  Q,  Q,  which  is  an  annihilator 
for  all  vectors  perpendicular  to  ft,  may  be  added  to  the  nu- 
merator. The  versor  0  may  then  be  written 

^=aa  +  (i  +  ixtt)* 

1  +  ft-ft 

(i  +  1  x  ft)  .  (i  +  1  x  a)  =  i  +  2  1  x  ft  +  (i  x  ft).  (i  x  Q) 
(i  x  ft)  •  (i  x  ft)  =  (i  x  ft)  x  ft  =  i  .  ft  ft  -  ft  .  ft  i. 

Hence  substituting  : 


1  +  ft.ft 

This  may  be  expanded  in  nonion  form.     Let 


0  = 


-(11) 


128.  ]     If  a  is  a  unit  vector  a  dyadic  of  the  form 

*  =  2aa-I  (12) 

is  a  Mquadrantal  versor.  That  is,  the  dyadic  0  turns  the 
points  of  space  about  the  axis  a  through  two  right  angles. 
This  may  be  seen  by  setting  q  equal  to  TT  in  the  general 
expression  for  a  versor 

0  =  a  a  +  cos  q  (I  —  a  a)  +  sin  £  I  x  a, 

or  it  may  be  seen  directly  from  geometrical  considerations. 
The  dyadic  0  leaves  a  vector  parallel  to  a  unchanged  but  re- 
verses every  vector  perpendicular  to  a  in  direction. 

Theorem :  The  product  of  two  biquadrantal  versors  is  a 
versor  the  axis  of  which  is  perpendicular  to  the  axes  of  the 


344  VECTOR  ANALYSIS 

biquadrantal  versors  and    the  angle    of  which  is  twice  the 
angle  from  the  axis  of  the  second  to  the  axis  of  the  first. 
Let  a  and  b  be  the  axes  of  two  biquadrantal  versors.     The 

product 

£=(2bb-I).(2aa-I) 

is  certainly  a  versor;  for  the  product  of  any  two  versors 
is  a  versor.  Consider  the  common  perpendicular  to  a  and  b. 
The  biquadrantal  versor  2  a  a  —  I  reverses  this  perpendicular 
in  direction.  (2bb—  I)  again  reverses  it  in  direction  and  con- 
sequently brings  it  back  to  its  original  position.  Hence  the 
product  Q  leaves  the  common  perpendicular  to  a  and  b  un- 
changed. Q  is  therefore  a  rotation  about  this  line  as  axis. 


The  cosine  of  the  angle  from  a  to  Q  •  a  is 
a  •  Q  .  a  =  2  b  •  a  b  •  a  —  a  .  a  =  2  (b  .  a)2  -  1  =  cos  2  (b,  a). 

Hence  the  angle  of  the  versor  Q  is  equal  to  twice  the  angle 
from  a  to  b. 

Theorem  :  Conversely  any  given  versor  may  be  expressed 
as  the  product  of  two  biquadrantal  versors,  of  which  the  axes 
lie  in  the  plane  perpendicular  to  the  axis  of  the  given  versor 
and  include  between  them  an  angle  equal  to  one  half  the 
angle  of  the  given  versor. 

For  let  Q  be  the  given  versor.  Let  a  and  b  be  unit  vectors 
perpendicular  to  the  axis  —  J?x  of  this  versor.  Furthermore 
let  the  angle  from  a  to  b  be  equal  to  one  half  the  angle  of 
this  versor.  Then  by  the  foregoing  theorem 

Q  =  (2bb  -  I)  .  (2  aa  -  I).  (14) 

The  resolution  of  versors  into  the  product  of  two  biquad- 
rantal versors  affords  an  immediate  and  simple  method  for 
compounding  two  finite  rotations  about  a  fixed  point  Let 
0  and  W  be  two  given  versors.  Let  b  be  a  unit  vector  per- 


ROTATIONS  AND  STRAINS  345 

pendicular  to  the  axes  of  0  and  ¥.  Let  a  be  a  unit  vector 
perpendicular  to  the  axis  of  0  and  such  that  the  angle  from 
a  to  b  is  equal  to  one  half  the  angle  of  0.  Let  c  be  a  unit 
vector  perpendicular  to  the  axis  of  W  and  such  that  the  angle 
from  b  to  c  is  equal  to  one  half  the  angle  of  W.  Then 

<P  =  (2bb-I).(2aa-I) 


?F.  0  =  (2  c  c  -  I)  •  (2  b  b  -  I)2  .  (2  a  a  -  I). 

But  (2  bb  —  I)2  is  equal  to  the  idemfactor,  as  may  be  seen  from 
the  fact  that  it  represents  a  rotation  through  four  right  angles 
or  from  the  expansion 

(2  b  b  -  I)  •  (2  b  b  -  I)  =  4  b  •  b  bb-4bb  +  I  =  I. 

Hence  ¥•  0  =  (2  cc  -  I)  .(2  aa  -  I). 

The  product  of  W  into  0  is  a  versor  the  axis  of  which  is 
perpendicular  to  a  and  c  and  the  angle  of  which  is  equal  to 
one  half  the  angle  from  a  to  c. 

If  0  and  W  are  two  versors  of  which  the  vector  semi- 
tangents  of  version  are  respectively  ftt  and  Q.J,  the  vector 
semi-tangent  of  version  Q3  of  the  product  W  •  <P  is 

q1  +  a2+Q2xa1  15 

i-ci^a, 

Let  d>  =  (2  bb  -  I)  .  (2  aa  -  I) 

and  V=  (2  cc  -  I)  •  (2  bb  -  I). 

W  •  0  =  (2cc-I)  •  (2aa-I). 


«*     — 


ba  —  2  a  a  —  2  b  b  +  I, 
=  4  a  •  b  b  x  a, 


346  VECTOR  ANALYSIS 

0s  =  4(a.b)2-l, 
F  =  4  c  •  b   cb   -2bb  -2cc   +  I, 
?FX  =  4  c  «  b   c  x  b, 

?F  •  <P  =  4  c  •  a  ca  —  2  c  c   —  2  a  a   +  I, 

(  ¥  •  $)x  =  4  c  •  a  c  x  a, 
(?F  •  0)^  —  4  (c  •  a)2  —  1. 
axb  bxc  axe 

HfillCfi  QJ     ~^r  - Q.    — ~~ Q    • — *  • 

(b  X  c)  x  (a  x  b)          fa  b  c]  b 

O       y    Q      —  — —  —  — 

Ho   A   Hi  —  —  r     —  ,    ,  • 

ft  •  D  b  •  0  a  -b  b  •  c 

But          [a be]  r  =  bxc  a»r  +  cxa  b»r  +  axb  c»r, 
Hence      [abc]b  =  bxc  a«b  +  cxab«b  +  axbc.b. 

bxc      axb         axe 

b  •  c         a  •  b       a  •  b  b  •  c 

a  •  c  ftQ 

Hence  ft2  x  ft,  =  —  Q,,  —  ft2  H —~- 

a •  bb»  c 


a.e 


a  •  b  b>  c 

_(axb)»(bxc)_a«bb.c      a  •  c  b  •  b 

a  •  b  b  •  c  a  •  b  b  •  c      a  •  b  b  •  c 


Hence  Q       Q!  x  g2  +  ft2  +  ft, 


ROTATIONS  AND  STRAINS  347 

This  formula  gives  the  composition  of  two  finite  rotations. 
If  the  rotations  be  infinitesimal  ftj  and  Q,2  are  both  infinitesi- 
mal. Neglecting  infinitesimals  of  the  second  order  the  for- 

mula reduces  to 

Q,3  =  Q,j  +  Q,2. 

The  infinitesimal  rotations  combine  according  to  the  law  of 
vector  addition.  This  demonstrates  the  parallelogram  law  for 
angular  velocities.  The  subject  was  treated  from  different 
standpoints  in  Arts.  51  and  60. 

Cyclics,  Eight  Tensors,  Tonics,  and  Cyclotonics 

129.]  If  the  dyadic  <P  be  a  versor  it  may  be  written  in  the 
form  (4) 

0  =  i  i  +  cos  q  (j  j  +  k  k)  +  sin  q  (k  j  —  j  k). 

The  axis  of  rotation  is  i  and  the  angle  of  rotation  about  that 
axis  is  q.  Let  W  be  another  versor  with  the  same  axis  and 
an  angle  of  rotation  equal  to  q'. 

W  =  i  i  +  cos  q'  (j  j  +  k  k)  +  sin  q'  (k  j  —  j  k). 

Multiplying  : 

0  .  W  =  W  •  0  =  i  i  cos  (2  +  ?')  (j  j  +  kk) 

kj-jk).  (16) 


This  is  the  result  which  was  to  be  expected  —  the  product  of 
two  versors  of  which  the  axes  are  coincident  is  a  versor  with 
the  same  axis  and  with  an  angle  equal  to  the  sum  of  the 
angles  of  the  two  given  versors. 

If  a  versor  be  multiplied  by  itself,  geometric  and  analytic 
considerations  alike  make  it  evident  that 

02  =  ii  +  cos  2q  (j  j  +  kk)  +  sin  2  q  (kj  -jk), 
and    0n  =  ii  +  cos  nq  (jj  +  kk)  +  sin  nq  (kj  —  j  k). 


348  VECTOR  ANALYSIS 

On  the  other  hand  let  0l  equal  jj  +  kk;  and  02  equal 
kj-jk.  Then 

4>*  —  (i  i  +  cos  q  &!  +  sin  q  $2)n. 

The  product  of  ii  into  either  0l  or  <P2  is  zero  and  into  itself  is 

i  i.     Hence 

Qn  =  ii  +  (cos  q  <Pl  +  sin  q  02)n 

0n=  ii  +  cosn<?  $!*  +  w  cos""1*?  sin  ^  (Pj"-1  .  02  +  •  •  . 

The  dyadic  </>1  raised  to  any  power  reproduces  itself.  0^  —  <Pr 
The  dyadic  02  raised  to  the  second  power  gives  the  negative 
of  $!  ;  raised  to  the  third  power,  the  negative  of  <PZ  ;  raised 
to  the  fourth  power,  01  ;  raised  to  the  fifth  power,  <P2  and  so 
on  (Art.  114).  The  dyadic  01  multiplied  by  $2  is  equal  to 
02.  Hence 

0n  =  ii  +  cosn  q  01  +  n  cosn~1q  sin  q  <PZ 

n(n—~L) 
-       2!       cos"-2  q  sm2  $2  +  .  •  • 

But  0"  =  i  i  +  cos  n  q  01  +  sin  71  q  <$v 

Equating  coefficients  of  01  and  tf>2  in  these  two  expressions 
for  <Pn 

n  (n  —  1) 
cos  n  q  =  cos*  £  --  ^-^-.  -  -  cos""2  q  sm2  q  +  •  •  • 

A  I 


• 

Thus  the  ordinary  expansions  for  cos  nq  and  sinnq  are 
obtained  in  a  manner  very  similar  to  the  manner  in  which 
they  are  generally  obtained. 

The  expression  for  a  versor  may  be  generalized  as  follows. 
Let  a,  b,  c  be  any  three  non-coplanar  vectors  ;  and  a',  V,  c',  the 
reciprocal  system.  Consider  the  dyadic 

cc')  +  sin  q  (cb'-bc')-  (IT) 


ROTATIONS  AND  STRAINS  349 

This  dyadic  leaves  vectors  parallel  to  a  unchanged.  Vectors 
in  the  plane  of  b  and  c  suffer  a  change  similar  to  rotation. 

Let 

r  =  cos  p  b  +  sin  p  c, 

r'  =  0  .  r  =  cos  (j?  +  q)  b  +  sin  (p  +  q)  c. 

This  transformation  may  be  given  a  definite  geometrical 
interpretation  as  follows.  The  vector  r,  when  p  is  regarded 
as  a  variable  scalar  parameter,  describes  an  ellipse  of  which 
b  and  c  are  two  conjugate  semi-diameters  (page  117).  Let 
this  ellipse  be  regarded  as  the  parallel  projection  of  the 

unit  circle 

f  =  cos  p  i  +  sin  q  j. 

That  is,  the  ellipse  and  the  circle  are  cut  from  the  same 
cylinder.  The  two  semi-diameters  i  and  j  of  the  circle  pro- 
ject into  the  conjugate  semi-diameters  a  and  b  of  the  ellipse. 
The  radius  vector  r  in  the  ellipse  projects  into  the  radius  vector 
f  in  the  unit  circle.  The  radius  vector  r'  in  the  ellipse  which 
is  equal  to  <#•  r,  projects  into  a  radius  vector  r'  in  the  circle 

such  that 

f '  =  cos  (p  +  q)  i  +  sin  (jp  +  q)  j. 

Thus  the  vector  r  in  the  ellipse  is  so  changed  by  the  applica- 
tion of  0  as  a  prefactor  that  its  projection  f  in  the  unit  circle 
is  rotated  through  an  angle  q. 

This  statement  may  be  given  a  neater  form  by  making  use 
of  the  fact  that  in  parallel  projection  areas  are  changed  in  a 
definite  constant  ratio.  The  vector  f  in  the  unit  circle  may 
be  regarded  as  describing  a  sector  of  which  the  area  is  to  the 
area  of  the  whole  circle  as  q  is  to  2  TT.  The  radius  vector  f 
then  describes  a  sector  of  the  ellipse.  The  area  of  this  sector 
is  to  the  area  of  the  whole  ellipse  as  q  is  to  2  TT.  Hence  the 
dyadic  0  applied  as  a  prefactor  to  a  radius  vector  r  in  an  ellipse 
of  which  b  and  c  are  two  conjugate  semi-diameters  advances 
that  vector  through  a  sector  the  area  of  which  is  to  the  area  of 


350  VECTOR  ANALYSIS 

the  whole  ellipse  as  q  is  to  2-Tr.1  Such  a  displacement  of  the 
radius  vector  r  may  be  called  an  elliptic  rotation  through  a 
sector  q  from  its  similarity  to  an  ordinary  rotation  of  which 
it  is  the  projection. 

Definition  :  A  dyadic  0  of  the  form 

0  =  a  a'  +  cos  q  (b  V  4-  c  c')  +  sin  q  (c  V  —  b  c')    (17) 

is  called  a  cyclic  dyadic.  The  versor  is  a  special  case  of  a 
cyclic  dyadic. 

It  is  evident  from  geometric  or  analytic  considerations  that 
the  powers  of  a  cyclic  dyadic  are  formed,  as  the  powers  of  a 
versor  were  formed,  by  multiplying  the  scalar  q  by  the  power 
to  which  the  dyadic  is  to  be  raised. 

0n  —  a  a'  +  cos  nq  (b  V  +  c  c')  +  sin  nq  (c  b'  —  b  c'). 
If  the  scalar  q  is  an  integral  sub-multiple  of  2  TT,  that  is,  if 

2?r 

—  =  m, 

9. 

it  is  possible  to  raise  the  dyadic  0  to  such  an  integral  power, 
namely,  the  power  m,  that  it  becomes  the  idemfactor 


0  may  then  be  regarded  as  the  rath  root  of  the  idemfactor. 
In  like  manner  if  q  and  2  TT  are  commensurable  it  is  possible 
to  raise  0  to  such  a  power  that  it  becomes  equal  to  the  idem- 
factor  and  even  if  q  and  2  TT  are  incommensurable  a  power  of 
0  may  be  found  which  differs  by  as  little  as  one  pleases  from 
the  idemfactor.  Hence  any  cyclic  dyadic  may  be  regarded  as 
a  root  of  the  idemfactor. 

1  It  is  evident  that  fixing  the  result  of  the  application  of  4>  to  all  radii  vectors 
in  an  ellipse  practically  fixes  it  for  all  vectors  in  the  plane  of  b  and  c.  For  any 
vector  in  that  plane  may  be  regarded  as  a  scalar  multiple  of  a  radius  vector  of 
the  ellipse. 


ROTATIONS  AND  STRAINS  351 

130.]  Definition:  The  transformation  represented  by  the 
d?adic  0  =  aii  +  Jjj+Ckk  (18) 

where  a,  &,  c  are  positive  scalars  is  called  a  pure  strain.  The 
dyadic  itself  is  called  a  right  tensor. 

A  right  tensor  may  be  factored  into  three  factors 

0=  (aii+jj  +  kk>(-(ii  +  &j  j  +  kk)  +(ii+jj  +  ckk). 

The  order  in  which  these  factors  occur  is  immaterial.  The 
transformation 

p'  =  (ii  +  jj  +  ckk)  .  r 

is  such  that  the  i  and  j  components  of  a  vector  remain  un- 
altered but  the  k-component  is  altered  in  the  ratio  of  c  to  1. 
The  transformation  may  therefore  be  described  as  a  stretch  or 
elongation  along  the  direction  k.  If  the  constant  c  is  greater 
than  unity  the  elongation  is  a  true  elongation :  but  if  c  is  less 
than  unity  the  elongation  is  really  a  compression,  for  the  ratio 
of  elongation  is  less  than  unity.  Between  these  two  cases 
comes  the  case  in  which  the  constant  is  unity.  The  lengths 
of  the  k-components  are  then  not  altered. 

The  transformation  due  to  the  dyadic  0  may  be  regarded 
as  the  successive  or  simultaneous  elongation  of  the  com- 
ponents of  r  parallel  to  i,  j,  and  k  respectively  in  the  ratios 
a  to  1,  b  to  1,  c  to  1.  If  one  or  more  of  the  constants  a,  &,  c 
is  less  than  unity  the  elongation  in  that  or  those  directions 
becomes  a  compression.  If  one  or  more  of  the  constants  is 
unity,  components  parallel  to  that  direction  are  not  altered. 
The  directions  i,  j,  k  are  called  the  principal  axes  of  the  strain. 
Their  directions  are  not  altered  by  the  strain  whereas,  if  the 
constants  a,  &,  c  be  different,  every  other  direction  is  altered. 
The  scalars  a,  &,  c  are  known  as  the  principal  ratios  of 
elongation. 

In  Art.  115  it  was  seen  that  any  complete  dyadic  was 
reducible  to  the  normal  form 

0=  ±  (ai'i  +  Jj'j  +  ck'k)' 


352  VECTOR   ANALYSIS 

where  a,  Z>,  c  are  positive  constants.  This  expression  may  be 
factored  into  the  product  of  two  dyadics. 

0  =  ±  (a  i/i/  +  &  j'j'  +  c  k'k')  .  (i'i+j'j  +  k'k),     (19) 
or        0=  ±  (i'i  +j'j  +  k'k)  •  (a  ii  +  6jj  +  ckk). 
The  factor  i'i+j'j  +  k'k 

which  is  the  same  in  either  method  of  factoring  is  a  versor. 
It  turns  the  vectors  i,  j,  k  into  the  vectors  i ',  j ',  k'.  This 
versor  may  be  represented  by  its  vector  semi-tangent  of 
version  as 

•/•       i/j       v/  TT  _  i  x  *'  +  J  x  ^  +  k  x  k/ 
"l  +  i.i'+j.j'  +  k.k'' 

The  other  factor 

a  i'i' +  6  j'j'  +  ck'k', 

or  a  i  i  - 

is  a  right  tensor  and  represents  a  pure  strain.  In  the  first 
case  the  strain  has  the  lines  i',  j',  k'  for  principal  axes:  in 
the  second,  i,  j,  k.  In  both  cases  the  ratios  of  elongation  are 
the  same,  —  a  to  1,  &  to  1,  c  to  1.  If  the  negative  sign  occurs 
before  the  product  the  version  and  pure  strain  must  have 
associated  with  them  a  reversal  of  directions  of  all  vectors  in 
space  —  that  is,  a  perversion.  Hence 

Theorem :  Any  dyadic  is  reducible  to  the  product  of  a 
versor  and  a  right  tensor  taken  in  either  order  and  a  positive 
or  negative  sign.  Hence  the  most  general  transformation 
representable  by  a  dyadic  consists  of  the  product  of  a  rota- 
tion or  version  about  a  definite  axis  through  a  definite  angle 
accompanied  by  a  pure  strain  either  with  or  without  perver- 
sion. The  rotation  and  strain  may  be  performed  in  either 
order.  In  the  two  cases  the  rotation  and  the  ratios  of  elonga- 
tion of  the  strain  are  the  same ;  but  the  principal  axes  of  the 
strain  differ  according  as  it  is  performed  before  or  after  the 


ROTATIONS  AND  STRAINS  353 

rotation,  either  system  of  axes  being  derivable  from  the  other 
by  the  application  of  the  versor  as  a  prefactor  or  postfactor 
respectively. 

If  a  dyadic  0  be  given  the  product  of  0  and  its  conjugate 
is  a  right  tensor  the  ratios  of  elongation  of  which  are  the 
squares  of  the  ratios  of  elongation  of  0  and  the  axes  of  which 
are  respectively  the  antecedents  or  consequents  of  0  accord- 
ing as  0C  follows  or  precedes  0  in  the  product. 

d>=  ±  (ai'i  +  6  j'j  +  ck'k), 
0C  =  ±  (aii'  +  J  jj'  +  ckk'), 
0.  0tf=aai'i'  +  &2j'j'  +  cak'k',  (20) 

0C.  0  =  a*ii  +  Z>2jj  +  c2kk. 

The  general  problem  of  finding  the  principal  ratios  of  elonga- 
tion, the  antecedents,  and  consequents  of  a  dyadic  in  its 
normal  form,  therefore  reduces  to  the  simpler  problem  of  find- 
ing the  principal  ratios  of  elongation  and  the  principal  axes 
of  a  pure  strain. 

131.]     The  natural  and  immediate  generalization  of   the 

right  tensor 

ckk, 


is  the  dyadic         0  =  a  aa'  4-  &  b  b'  +  c  c  c'  (21) 

where  a,  6,  c  are  positive  or  negative  scalars  and  where  a,  b,  c 
and  a',  b',  c'  are  two  reciprocal  systems  of  vectors.     Neces- 
sarily a,  b,  c  and  a',  b',  c'  are  each  three  non-coplanar. 
Definition  :  A  dyadic  that  may  be  reduced  to  the  form 


ccc'  (21) 

is  called  a  tonic. 

The  effect  of  a  tonic  is  to  leave  unchanged  three  non- 
coplanar  directions  a,  b,  c  in  space.  If  a  vector  be  resolved 
into  its  components  parallel  to  a,  b,  c  respectively  these 

23 


354  VECTOR  ANALYSIS 

components  are  stretched  in  the  ratios  a  to  1,  &  to  1,  c  to  1. 
If  one  or  more  of  the  constants  a,  6,  c  are  negative  the  com- 
ponents parallel  to  the  corresponding  vector  a,  b,  c  are  re- 
versed in  direction  as  well  as  changed  in  magnitude.  The 
tonic  may  be  factored  into  three  factors  of  which  each 
stretches  the  components  parallel  to  one  of  the  vectors  a,  b,  c 
but  leaves  unchanged  the  components  parallel  to  the  other 
two. 


The  value  of  a  tonic  0  is  not  altered  if  in  place  of  a,  b,  c 
any  three  vectors  respectively  collinear  with  them  be  sub- 
stituted, provided  of  course  that  the  corresponding  changes 
which  are  necessary  be  made  in  the  reciprocal  system  a',  b',  c'. 
But  with  the  exception  of  this  change,  a  dyadic  which  is 
expressible  in  the  form  of  a  tonic  is  so  expressible  in  only 
one  way  if  the  constants  a,  Z>,  c  are  different.  If  two  of  the 
constants  say  5  and  c  are  equal,  any  two  vectors  coplanar 
with  the  corresponding  vectors  b  and  c  may  be  substituted 
in  place  of  b  and  c.  If  all  the  constants  are  equal  the  tonic 
reduces  to  a  constant  multiple  of  the  idemfactor.  Any  three 
non-coplanar  vectors  may  be  taken  for  a,  b,  c. 

The  product  of  two  tonics  of  which  the  axes  a,  b,  c  are  the 
same  is  commutative  and  is  a  tonic  with  these  axes  and 
with  scalar  coefficients  equal  respectively  to  the  products  of 
the  corresponding  coefficients  of  the  two  dyadics. 


V  =  az  a  a'  +  &2  b  b'  +  c2  c  c' 
0.  ¥=  V.0  =  aiaza&'  +  J^bb'  +  ejCaCc'.       (22) 

The  generalization  of  the  cyclic  dyadic 

a  a'  +  cos  q  (b  V  +  c  c')  +  sin  q  (c  b'  -  b  c') 
is  0  =  aaa'  +  KbV  +  cc')  +  c  (cb'-W),      (23) 


ROTATIONS  AND  STRAINS  355 

where  a,  b,  c  are  three  non-coplanar  vectors  of  which  a  ',  b  ',  c  ' 
is  the  reciprocal  system  and  where  the  quantities  a,  J,  c,  are 
positive  or  negative  scalars.  This  dyadic  may  be  changed 
into  a  more  convenient  form  by  determining  the  positive 
scalar  p  and  the  positive  or  negative  scalar  q  (which  may 
always  be  chosen  between  the  limits  ±  TT)  so  that 

J  =  p  cos  q 
and  c=psinq.  (24) 

That  is,  p  =  + 


and  tan     2=.  (24)' 

Then 

(P  =  a  a  a'  +  p  cos  q  (b  V  +  c  c')  +  p  sin  q  (c  V  —  b  c').      (25) 

This  may  be  factored  into  the  product  of  three  dyadics 
0=  (aaa'  +  bb'  +  cc')  •  (aa'  +p  bb'  +pcc')  • 
{a  a'  +  cos  q  (b  b'  +  c  c')  +  sin  q  (c  V  —  b  c')}. 

The  order  of  these  factors  is  immaterial.  The  first  is  a  tonic 
which  leaves  unchanged  vectors  parallel  to  b  and  c  but 
stretches  those  parallel  to  a  in  the  ratio  of  a  to  1.  If  a  is 
negative  the  stretching  must  be  accompanied  by  reversal 
in  direction.  The  second  factor  is  also  a  tonic.  It  leaves 
unchanged  vectors  parallel  to  a  but  stretches  all  vectors  in 
the  plane  of  b  and  c  in  the  ratio  p  to  1.  The  third  is  a 
cyclic  factor.  Vectors  parallel  to  a  remain  unchanged  ;  but 
radii  vectors  in  the  ellipse  of  which  b  and  c  are  conjugate 
semi-diameters  are  rotated  through  a  vector  such  that  the 
area  of  the  vector  is  to  the  area  of  the  whole  ellipse  as  q  to 
2  TT.  Other  vectors  in  the  plane  of  b  and  c  may  be  regarded 
as  scalar  multiples  of  the  radii  vectors  of  the  ellipse. 


356  VECTOR  ANALYSIS 

Definition :  A  dyadic  which  is  reducible  to  the  form 
<P  =  a  a  a'  +  p  cos  q  (b  V  +  c  c')  +  p  sin  q  (c  V  —  b  c'),      (25) 

owing  to  the  fact  that  it  combines  the  properties  of  the 
cyclic  dyadic  and  the  tonic  is  called  a  cyclotonic. 

The  product  of  two  cyclotonics  which  have  the  same  three 
vectors,  a,  b,  c  as  antecedents  and  the  reciprocal  system 
a',  b',  c'  for  consequents  is  a  third  cyclotonic  and  is  com- 
mutative. 

0  =  al  a  a'  +  p1  cos  ^  (b  V  +  c  c')  +  pl  sin  q1  (c  b'  —  b  c') 
¥  =  az  a  a'  +  p2  cos  q2  (b  V  +  c  c')  +  p2  sin  qz  (c  b'  —  b  c') 
0  .  W  =  ¥ •  d>  =  ax  a2  aa'  +  p^p2  cos  (q1  +  qz)  (bb'  +  c  c') 
+  Pi  PZ  sin  (ft  +  2a)  (c  b'  -  b  o1)-  (26) 

Reduction  of  Dyadics  to  Canonical  Forms 

132.]  Theorem :  In  general  any  dyadic  0  may  be  reduced 
either  to  a  tonic  or  to  a  cyclotonic.  The  dyadics  for  which 
the  reduction  is  impossible  may  be  regarded  as  limiting  cases 
which  may  be  represented  to  any  desired  degree  of  approxi- 
mation by  tonics  or  cyclotonics. 

From  this  theorem  the  importance  of  the  tonic  and  cyclo- 
tonic which  have  been  treated  as  natural  generalizations  of 
the  right  tensor  and  the  cyclic  dyadic  may  be  seen.  The 
proof  of  the  theorem,  including  a  discussion  of  all  the 
special  cases  that  may  arise,  is  long  and  somewhat  tedious. 
The  method  of  proving  the  theorem  in  general  however  is 
patent.  If  three  directions  a,  b,  c  may  be  found  which  are 
left  unchanged  by  the  application  of  0  then  0  must  be  a 
tonic.  If  only  one  such  direction  can  be  found,  there  exists 
a  plane  in  which  the  vectors  suffer  a  change  such  as  that  due 
to  the  cyclotonic  and  the  dyadic  indeed  proves  to  be  such. 


ROTATIONS  AND  STRAINS  357 

The  question  is  to  find  the  directions  which  are  unchanged 
by  the  application  of  the  dyadic  0. 
If  the  direction  a  is  unchanged,  then 

d>  •  a  =  a  a  (27) 

or  (0  —  al).a  =  0. 

The  dyadic  0  —  a  I  is  therefore  planar  since  it  reduces  vectors 
in  the  direction  a  to  zero.  In  special  cases,  which  are  set 
aside  for  the  present,  the  dyadic  may  be  linear  or  zero.  In 
any  case  if  the  dyadic 

0-al 

reduces  vectors  collinear  with  a  to  zero  it  possesses  at  least 
one  degree  of  nullity  and  the  third  or  determinant  of  <P 
vanishes. 

(0-aI)8  =  0.  (28) 

Now  (page  331)  (3>  +  ¥)3  =  <P3  +  <P2  :  W  +  d>  :  ¥2  +  ¥3. 
Hence  (4>  -  al)3  =  <P8  -  a  &2  :  1  +  a2  0  :  12  -  a3  18 

I,  =  I  and  I3  =  1. 
But  0  :  1  =  ffi 


Hence  the  equation  becomes 

a3  -  a2  ffia  +  a  4>zS  -4>z  =  0.  (29) 

The  value  of  a  which  satisfies  the  condition  that 
0  •  a  =  aa 

is  a  solution  of  a  cubic  equation.     Let  x  replace  a.     The 
cubic  equation  becomes 

(29)' 


358  VECTOR  ANALYSIS 

Any  value  of  x  which  satisfies  this  equation  will  be  such 

that 

(0-sI)8  =  0.  (28)' 

That  is  to  say,  the  dyadic  0  —  x  I  is  planar.  A  vector  per- 
pendicular to  its  consequents  is  reduced  to  zero.  Hence  0 
leaves  such  a  direction  unchanged.  The  further  discussion 
of  the  reduction  of  a  dyadic  to  the  form  of  a  tonic  or  a  cycle- 
tonic  depends  merely  upon  whether  the  cubic  equation  in  x 
has  one  or  three  real  roots. 

133.  J     Theorem  :  If  the  cubic  equation 

xs  -  x*  03  +  x  &Z3  -0S=0  (29)' 

has  three  real  roots  the  dyadic  0  may  in  general  be  reduced 
to  a  tonic. 

For  let  x  =  a,     x  =  &,     x  =  c 

be  the  three  roots  of  the  equation.     The  dyadics 
0  —  a  I,     <P  —  bl,     0  —  cl 

are  in  general  planar.  Let  a,  b,  c  be  respectively  three 
vectors  drawn  perpendicular  to  the  planes  of  the  consequents 
of  these  dyadics. 


b  =  0,  (30) 

(0-cI).c  =  0. 
Then  $  .  a  =  a  a, 

0.b  =  Jb,  (30)' 

0  •  C  =  C  C. 

If  the  roots  a,  Z>,  c  are  distinct  the  vectors  a,  b,  c  are  non- 
coplanar.     For  suppose 

c  =  ma  +  Ttb 


ROTATIONS  AND   STRAINS  359 

m  (P  •  a,  —  m  c  a  +  n  <P  •})  —  ?i  c  b  =  0. 
But  0.a  =  «a,     <P.b=&b. 

Hence  w  (a  —  c)  a  +  ft  (&  —  c)  b  =  0, 

and  m  (a  —  c)  =  0,     w  (b  —  c)  =  0. 

Hence  m  =  0  or  a  =  c,    n  =  0  or  b  =  c. 

Consequently  if  the  vectors  a,  b,  c  are  coplanar,  the  roots  are 
not  distinct;  and  therefore  if  the  roots  are  distinct,  the 
vectors  a,  b,  c  are  necessarily  non-coplanar.  In  case  the  roots 
are  not  distinct  it  is  still  always  possible  to  choose  three 
non-coplanar  vectors  a,  b,  c  in  such  a  manner  that  the  equa- 
tions (30)  hold.  This  being  so,  there  exists  a  system  a',  b',  c' 
reciprocal  to  a,  b,  c  and  the  dyadic  which  carries  a,  b,  c  into 
a  a,  b  b,  c  c  is  the  tonic 


ccc. 

Theorem  :  If  the  cubic  equation 

xs  -  x2  tf>s  +  x  <P2S  -  <P3  =  0  (29)' 

has  one  real  root  the  dyadic  0  may  in  general  be  reduced  to 
a  cyclotonic. 

The  cubic  equation  has  one  real  root.  This  must  be  posi- 
tive or  negative  according  as  <P8  is  positive  or  negative.  Let 
the  root  be  a.  Determine  a  perpendicular  to  the  plane  of 
the  consequents  of  0  —  a  I. 

(0-aI).a  =  0, 

Determine  a'  also  so  that 


and  let  the  lengths  of  a  and  a'  be  so  adjusted  that  a'  •  a  =  l. 
This  cannot  be  accomplished  in  the  special  case  in  which  a 


360  VECTOR  ANALYSIS 

and  a'  are  mutually  perpendicular.  Let  b  be  any  vector  in 
the  plane  perpendicular  to  a'. 

a'  •  (<P  -  a  I)  •  b  =  0. 

Hence  (0—  al)»b  is  perpendicular  to  a'.  Hence  0»b  is 
perpendicular  to  a'.  In  a  similar  manner  $2«  b,  <03«b,  and 
0"1  •  b,  CP"2  •  b,  etc.,  will  all  be  perpendicular  to  a'  and  lie  in 
one  plane.  The  vectors  <P  •  b  and  b  cannot  be  parallel  or  0 
would  have  the  direction  b  as  well  as  a  unchanged  and 
thus  the  cubic  would  have  more  than  one  real  root. 

The  dyadic  0  changes  a,  0  •  b,  b  into  4>  •  a,  <PZ  •  b,  <P  •  b  re- 
spectively.    The  volume  of  the  parallelepiped 

[<P  .  a     02  •  b     0  •  b]  =  <P8  [a     0  •  b    b].          (31) 
But  0  •  a  =  a  a. 

Hence     a  a  •  (<P2  •  b)  x  (®  •  b)  =  tf>3  a  •  (<P  .  b)  x  b.       (31)' 

The  vectors  <02  •  b,  0  •  b,  b  all  lie  in  the  same  plane.  Their 
vector  products  are  parallel  to  a'  and  to  each  other.  Hence 


b)  x  (0-b)  =  08     0«bxb.  (31)" 

Inasmuch  as  a  and  <P3  have  the  same  sign,  let 

*»  =  a-i08.  (32) 

Let  also  b3  =  p  ~l  4>  •  b        b2  =p~*  $*  •  b,    etc.     (33) 

and        b_1=^2^-1-b        b^  =^  0~2  •  b,     etc. 

b2  x  bj^bj  x  b2, 

or  (b2  +  b)  x  bj  =  0. 

The  vectors  b2  +  b  and  bx  are  parallel.     Let 

b2  +  b  =  2  n  br  (34) 

Then    b3  +  b1  =  2?ib2        b1  +  b2  =  27ib3         etc., 

(35) 
b   +  b_  =  2wb      b.  +  b_r=2wb_         etc. 


ROTATIONS  AND  STRAINS  361 

Lay  off  from  a  common  origin  the  vectors 

b,  bp  b2,  etc.,        b_j,  b_2,  etc. 

Since  0  is  not  a  tonic,  that  is,  since  there  is  no  direction  in 
the  plane  perpendicular  to  a'  which  is  left  unchanged  by  0 
these  vectors  bm  pass  round  and  round  the  origin  as  ra  takes 
on  all  positive  and  negative  values.  The  value  of  n  must 
therefore  lie  between  plus  one  and  minus  one.  Let 

n  =  cos  q.  (36) 

Then  b_j  +  b:  =  2  cos  q  b. 

Determine  c  from  the  equation 

bj  =  cos  q  b  +  sin  q  c. 
Then  b_j  =  cos  q  b  —  sin  q  c. 

Let  a',  b',  c'  be  the  reciprocal  system  of  a,  b,  c.  This  is  pos- 
sible since  a'  was  so  determined  that  a'  •  a  =  1  and  since 
a,  b,  c  are  non-coplanar.  Let 

W  =  cos  q  (bb'  +  cc')  +  sin  q  (c  V  -  be'). 
Then         W  .  a  =  0,         SF.b^bj,         3P  .  b^  =  b. 
Hence  (a  a,*'  +  p  ¥)  •  &  =  a  a  =  0  •  a, 

(aaa'  +  .p  ¥)  •  "b  =  p  b2  =  0  •  b, 
(a  aa'  +  p  ¥)  .  b_j  =p  b  =  <P  •  b_r 

The  dyadic  a  a.a!  +  p  ¥  changes  the  vectors  a,  b  and  b_j  into 
the  vectors  0  •  a,  0  •  b,  and  0  •  b_g  respectively.  Hence 

0  =  (a  a  a'  +  p  W)  =  a  a  a'  +  p  cos  0  (b  V  +  c  c') 

+  p  sin  q  (c  V  —  b  c'). 

The  dyadic  0  in  case  the  cubic  equation  has  only  one  real 
root  is  reducible  except  in  special  cases  to  a  cyclotonic. 
The  theorem  that  a  dyadic  in  general  is  reducible  to  a  tonic 
or  cyclotonic  has  therefore  been  demonstrated. 


362  VECTOR  ANALYSIS 

134.]  There  remain  two  cases1  in  which  the  reduction 
is  impossible,  as  can  be  seen  by  looking  over  the  proof.  In 
the  first  place  if  the  constant  n  used  in  the  reduction  to  cyclo- 
tonic form  be  ±  1  the  reduction  falls  through.  In  the  second 
place  if  the  plane  of  the  antecedents  of 

0-al 

and  the  plane  of  the  consequents  are  perpendicular  the 
vectors  a  and  a'  used  in  the  reduction  to  cyclotonic  form  are 
perpendicular  and  it  is  impossible  to  determine  a'  such  that 
a  •  a'  shall  be  unity.  The  reduction  falls  through. 

If  n  =  ±  1,        b_j  +  bx  =  ±  2  b. 

Let  b_i  +  bj  =  2  b. 

Choose  c^bj  —  b  =  b  —  b_r 

Consider  the  dyadic   ¥  =  a  a  a'  +  p  (b  b'  +  c  c')  -f-  p  c  b' 

¥  •  C  =p  G  =p'b1  —  J9  b  =  0  •  C. 

Hence  0  —  a  a  a'  +  p  (b  b'  +  c  c')  +  p  cbr          (37) 

The  transformation  due  to  this  dyadic  may  be  seen  best  by 
factoring  it  into  three  factors  which  are  independent  of  the 
order  or  arrangement 

0  =  (a  a  a'  +  b  b'  +  c  c')  •  {a  a/  +  p  (b  b'  +  c  c')} 

•  (a  a'  +  bb'  +  cc'  +  cb'). 

1  In  these  cases  it  will  be  seen  that  the  cubic  equation  has  three  real  roots. 
In  one  case  two  of  them  are  equal  and  in  the  other  case  three  of  them.  Thus 
these  dyadics  may  be  regarded  as  limiting  cases  lying  between  the  cyclotonic  in 
which  two  of  the  roots  are  imaginary  and  the  tonic  in  which  all  the  roots  are  real 
and  distinct.  The  limit  may  be  regarded  as  taking  place  either  by  the  pure 
imaginary  part  of  the  two  imaginary  roots  of  the  cyclotonic  becoming  zero  or  by 
two  of  the  roots  of  the  tonic  approaching  each  other. 


ROTATIONS  AND  STRAINS  363 

The  first  factor  represents  an  elongation  in  the  direction  a  in  a 
ratio  a  to  1.  The  plane  of  b  and  c  is  undisturbed.  The 
second  factor  represents  a  stretching  of  the  plane  of  b  and  c  in 
the  ratio  p  to  1.  The  last  factor  takes  the  form 

I  +  cb'. 

(I  +  c  V)  •  x&  =  #a, 

(I  +  c  b')  •  x  b  =  x  b  +  x  c, 

(I  +  c  V)  •  x  c  =  x  c. 

A  dyadic  of  the  form  I  +  cb'  leaves  vectors  parallel  to  a  and  c 
unaltered.  A  vector  x  b  parallel  to  b  is  increased  by  the  vec- 
tor c  multiplied  by  the  ratio  of  the  vector  x  b  to  b.  In  other 
words  the  transformation  of  points  in  space  is  such  that  the 
plane  of  a  and  c  remains  fixed  point  for  point  but  the  points 
in  planes  parallel  to  that  plane  are  shifted  in  the  direction  c 
by  an  amount  proportional  to  the  distance  of  the  plane  in 
which  they  lie  from  the  plane  of  a  and  c. 
Definition  :  A  dyadic  reducible  to  the  form 

I  +  cb' 

is  called  a  shearing  dyadic  or  shearer  and  the  geometrical 
transformation  which  it  causes  is  called  a  shear.  The  more 
general  dyadic 

cc')  +  oV  (37) 


will  also  be  called  a  shearing  dyadic  or  shearer.  The  trans- 
formation to  which  it  gives  rise  is  a  shear  combined  with 
elongations  in  the  direction  of  a  and  is  in  the  plane  of  b  and  c. 
If  n  =  —1  instead  of  n  =  +1,  the  result  is  much  the  same. 
The  dyadic  then  becomes 


cc')-cV  (37)' 

0  =  (a  aa'  +  bV  +  cc')  .  {aa'  -p  (bV  +  cc')}  •  (I  +  cV). 


364  VECTOR  ANALYSIS 

The  factors  are  the  same  except  the  second  which  now  repre- 
sents a  stretching  of  the  plane  of  b  and  c  combined  with  a 
reversal  of  all  the  vectors  in  that  plane.  The  shearing  dyadic 
0  then  represents  an  elongation  in  the  direction  a,  an  elonga- 
tion combined  with  a  reversal  of  direction  in  the  plane  of 
b  and  c,  and  a  shear. 

Suppose  that  the  plane  of  the  antecedents  and  the  plane  of 
the  consequents  of  the  dyadic  <#—  al  are  perpendicular.  Let 
these  planes  be  taken  respectively  as  the  plane  of  j  and  k  and 
the  plane  of  i  and  j  k.  The  dyadic  then  takes  the  form 


The  coefficient  B  must  vanish.     For  otherwise  the  dyadic 


is  planar  and  the  scalar  a  +  B  is  a  root  of  the  cubic  equation. 
With  this  root  the  reduction  to  the  form  of  a  tonic  may  be 
carried  on  as  before.  Nothing  new  arises.  But  if  B  vanishes 
a  new  case  occurs.  Let 


This  may  be  reduced  as  follows  to  the  form 

ab'  +  be' 

where      a  •  V  =  a  •  e'  =  b  •  e'  =  0        and  b  •  b'  =  1. 
Square  V  W*  =  A  D  ki  =  ac'. 

Hence  a  must  be  chosen  parallel  to  k  ;  and  c',  parallel  to  i. 
The  dyadic  ¥  may  then  be  transformed  into 


Then  =  A  D  k,        b'  =  Ci+Di 

A  D 


O  = 


ROTATIONS  AND  STRAINS  365 

With  this  choice  of  a,  b,  V,  c'  the  dyadic  W  reduces  to  the 
desired  form  ab'+  be'  and  hence  the  dyadic  0  is  reduced  to 

0  =  al  +  ab'  +  bc'  (38) 

or  0  =  aaa'  +  abb'+  ace'  +  aV  +  be'. 

This  may  be  factored  into  the  product  of  two  dyadics  the 
order  of  which  is  immaterial. 


The  first  factor  al  represents  a  stretching  of  space  in  all 
directions  in  the  ratio  a  to  1.     The  second  factor 


represents  what  may  be  called  a  complex  shear.     For 
r'=  IT  +  ab'.r  +  bc'^r  =  r  +  ab'.r  +  bc'«  r. 

If  r  is  parallel  to  a  it  is  left  unaltered  by  the  dyadic  Q.  If 
r  is  parallel  to  b  it  is  changed  by  the  addition  of  a  term 
which  is  in  direction  equal  to  a  and  in  magnitude  propor- 
tional to  the  magnitude  of  the  vector  r.  In  like  manner 
if  r  is  parallel  to  c  it  is  changed  by  the  addition  of  a  term 
which  in  direction  is  equal  to  b  and  which  in  magnitude  is 
proportional  to  the  magnitude  of  the  vector  r. 

£.zb  =  (I  +  ab'  -f  bc')»  #b  =  #b  +  x& 
£»a;c  =  (I  +  ab'  +  bc')»#c  =  xc  +  #b. 

Definition  :  A  dyadic  which  may  be  reduced  to  the  form 
0  =  al  +  ab'  +  bc'  (38) 

is  called  a  complex  shearer. 

The  complex  shearer  as  well  as  the  simple  shearer  men- 
tioned before  are  limiting  cases  of  the  cyclotonic  and  tonic 
dyadics. 


366  VECTOR  ANALYSIS 

135.]  A  more  systematic  treatment  of  the  various  kinds 
of  dyadics  which  may  arise  may  be  given  by  means  of  the 
Hamilton-Cayley  equation 

03  _  0g  0z+  0zs  0_  03I  =  0  (39) 

and  the  cubic  equation  in  x 

x3—  <Paxz  +  <P23x—  03  =  0.  (29)' 

If  a,  &,  c  are  the  roots  of  this  cubic  the  Hamilton-Cayley 
equation  may  be  written  as 

(0-aI).(0-&I).(0-cI)  =0.         (40) 

If,  however,  the  cubic  has  only  one  root  the  Hamilton-Cayley 
equation  takes  the  form 


-  a  I).  (02  -2^?  cos  £  <P+p*I)  =0.       (41) 


In  general  the  Hamilton-Cayley  equation  which  is  an  equa- 
tion of  the  third  degree  in  0  is  the  equation  of  lowest  degree 
which  is  satisfied  by  (P.  In  general  therefore  one  of  the  above 
equations  and  the  corresponding  reductions  to  the  tonic  or 
cyclotonic  form  hold.  In  special  cases,  however,  the  dyadic 
0  may  satisfy  an  equation  of  lower  degree.  That  equation 
of  lowest  degree  which  may  be  satisfied  by  a  dyadic  is  called 
its  characteristic  equation.  The  following  possibilities  occur. 


II.  (<P-aI 

III.  (0-  a  I).(<P-&  I)2  =  0. 

IV.  (0-  a  !)•(</>-  61)  =0. 
V.  (<P-aI)3  =  0. 

VI.  (0-aI)2  =  0. 

VII.  0-aI    =  0. 


ROTATIONS  AND   STRAINS  367 

In  the  first  case  the  dyadic  is  a  tonic  and  may  be  reduced 

to  the  form 

0  =  aaa'  +  Jbb'  +  ccc'. 

In  the  second  case  the  dyadic  is  a  cyclotonic  and  may  be 
reduced  to  the  form 

0  =  aaa'  +  p  cos  q  (bb'  +  cc')  +  p  sin  q  (cV  —  be'). 

In  the  third  case  the  dyadic  is  a  simple  shearer  and  may  be 
reduced  to  the  form 

<P  =  aaa'  +  b  (bb' +  cc')  +  cb'. 

In  the  fourth  case  the  dyadic  is  again  a  tonic.  Two  of  the 
ratios  of  elongation  are  the  same.  The  following  reduction 
may  be  accomplished  in  an  infinite  number  of  ways. 

<P  =  aaa'  +  &  (bb'  +  cc'). 

In  the  fifth  case  the  dyadic  is  a  complex  shearer  and  may  be 

so  expressed  that 

0=  al  +  ab'  +  bc'. 

In  the  sixth  case  the  dyadic  is  again  a  simple  shearer  which 
may  be  reduced  to  the  form 

0  =  al  +  cb'=a(aa'  -4-bb'  +cc')  +  cV. 

In  the  seventh  case  the  dyadic  is  again  a  tonic  which  may  be 
reduced  in  a  doubly  infinite  number  of  ways  to  the  form 

<p  =  al  =  a  (a a'  +  bb'  +  cc'). 

These  seven  are  the  only  essentially  different  forms  which  a 
dyadic  may  take.  There  are  then  only  seven  really  different 
kinds  of  dyadics  —  three  tonics  in  which  the  ratios  of  elonga- 
tion are  all  different,  two  alike,  or  all  equal,  and  the  cyclo- 
tonic together  with  three  limiting  cases,  the  two  simple  and 
the  one  complex  shearer. 


368  VECTOR  ANALYSIS 

Summary  of  Chapter  VI 

The  transformation  due  to  a  dyadic  is  a  linear  homogeneou 
strain.     The  dyadic   itself   gives   the  transformation  of  the 
points  in  space.     The  second  of  the  dyadic  gives  the  trans- 
formation of  plane  areas.     The  third  of  the  dyadic  gives  the 
ratio  in  which  volumes  are  changed. 

r'=0«r,        s'=02»s,        v'=0av. 

The  necessary  and  sufficient  condition  that  a  dyadic  repre- 
sent a  rotation  about  a  definite  axis  is  that  it  be  reducible  to 

the  form 

4>  =  i'i  +  j'j  +  k'k  (1) 

or  that  d>  •  <PC  =  I     <PS  =  +  1  (2) 

or  that  0  •  <PC  =  I     <P3  >  0 

The  necessary  and  sufficient  condition  that  a  dyadic  repre- 
sent a  rotation  combined  with  a  transformation  of  reflection 
by  which  each  figure  is  replaced  by  one  symmetrical  to  it  is 
that 

0  =  -(i'i  + j'  j  +  k'k)  (1)' 

or  that  <p.<pc  =  I,     d>3  =  —  1 

or  that  <P  •  <PC  =  I,     <P3  <  0.  (3) 

A  dyadic  of  the  form  (1)  is  called  a  versor ;  one  of  the  form 
(1)',  a  perversor. 

If  the  axis  of  rotation  of  a  versor  be  chosen  or  the  i-axis 
the  versor  reduces  to 

0  ==  ii  +  cos  q  (j  j  +  kk)  +  sin  q  (kj  -  j  k)      (4) 
or  0  =  i  i  +  cos  q  (I  —  i  i)  +  sin  q  I  x  i.  (5) 

If  any  unit  vector  a  is  directed  along  the  axis  of  rotation 

0  =  a  a  +  cos  q  (I  —  a  a)  +  sin  q  I  x  a         (6) 
The  axis  of  the  versor  coincides  in  direction  with  —  <PV. 


ROTATIONS  AND  STRAINS  369 

If  a  vector  be  drawn  along  the  axis  and  if  the  magnitude  of 
'ie  vector  be  taken  equal  to  the  tangent  of  one-half  the  angle 
£  rotation,  the  vector  determines  the  rotation  completely. 
fhis  vector  is  called  the  vector  semi-tangent  of  version. 

ft.  ft  =  tan2  ^2  (9) 

In  terms  of  ft  the  versor  0  may  be  expressed  in  a  number  of 
ways. 

ftft  /          ftft  \  Q 

&=-  —  +  cos  a  (I  —  r-  -  )  +  sin  q   I  x  —  =  —     (10) 

ft-ft  V      <J-<v  Va-a 

or  0  =  (I  +  I  x  a)  •  (I  -  I  x  ft)-1  (10)' 


•  (ioy 

1  +  ftft 


-. 

1  +  ft-ft 

If  a  is  a  unit  vector  a  dyadic  of  the  form 

0  =  2  a  a  -  I  (11) 

is  a  biquadrantal  versor.  Any  versor  may  be  resolved  into 
the  product  of  two  biquadrantal  versors  and  by  means  of 
such  resolutions  any  two  versors  may  be  combined  into 
another.  The  law  of  composition  for  the  vector  semi-tangents 
of  version  is 

ft     +  ft       +  ft     X  ft 


A  dyadic  reducible  to  the  form 
0  =  aa'  +  cos  q  (bV  +  cc')  +  sin  q  (cb'  —  be')       (17) 

is  called  a  cyclic  dyadic.     It  produces  a  generalization   of 
simple  rotation  —  an  elliptic  rotation,  so  to  speak.     The  pro- 

24 


370  VECTOR  ANALYSIS 

duct  of  two  cyclic  dyadics  which  have  the  same  antecedents 
a,  b,  c  and  consequents  a'  b'  c'  is  obtained  by  adding  their 
angles  q.  A  cyclic  dyadic  may  be  regarded  as  a  root  of  the 
idemfactor.  A  dyadic  reducible  to  the  form 

0  =  aii  +  &jj   +  ckk  (18) 

where  a,  &,  c  are  positive  scalars  is  called  a  right  tensor.  It 
represents  a  stretching  along  the  principal  axis  i,  j,  k  in  the 
ratio  a  to  1,  b  to  1,  c  to  1  which  are  called  the  principal  ratios 
of  elongation.  This  transformation  is  a  pure  strain. 

Any  dyadic  may  be  expressed  as  the  product  of  a  versor, 
a  right  tensor,  and  a  positive  or.  negative  sign. 

0  =  ±  (ai'i'  +  Jj'j'  +  c  k'k')  (i'i  +  j'j  +  k'k) 
or       0=  ±  (i'i  +  j'j  +  k'k).(aii+Jjj  +  ckk).        (19) 

Consequently  any  linear  homogeneous  strain  may  be  regarded 
as  a  combination  of  a  rotation  and  a  pure  strain  accompanied 
or  unaccompanied  by  a  perversion. 

The  immediate  generalizations  of  the  right  tensor  and  the 
cyclic  dyadic  is  to  the  tonic 


(21) 
and  cyclotonic 


cc')  +  c(cb'-bc)         (23) 
or    <P  =  aaa'  +  .p  cos  #  (bb'  +  cc')+^sin  #  (eb'-bc')    (25) 


where          p  =  +  */  52  .  C2  and  tan  I  a  =  -  -  .         (24)' 

p  +  b 

Any  dyadic  in  general  may  be  reduced  either  to  the  form 
(21),  and  is  therefore  a  tonic,  or  to  the  form  (25),  and  is 
therefore  a  cyclotonic.  The  condition  that  a  dyadic  be  a 
tonic  is  that  the  cubic  equation 

(29)' 


ROTATIONS  AND  STRAINS  371 

shall  have  three  real  roots.  Special  cases  in  which  the 
reduction  may  be  accomplished  in  more  ways  than  one  arise 
when  the  equation  has  equal  roots.  The  condition  that  a 
dyadic  be  a  cyclotonic  is  that  this  cubic  equation  shall  have 
only  one  real  root.  There  occur  two  limiting  cases  in  which 
the  dyadic  cannot  be  reduced  to  cyclotonic  form.  In  these 
cases  it  may  be  written  as 

0  =aaa'  +p  (bb'  +  cc')  +  cb'  (37) 

and  is  a  simple  shearer,  or  it  takes  the  form 

0  =  al  +  ab'  +  bc'  (38) 

and  is  a  complex  shearer.  Dyadics  may  be  classified  accord- 
ing to  their  characteristic  equations 

(<P  —  aI).(<P  —  61).  (<Z>  —  cl)  =0  tonic 

(0  —  a  I)  •  (<P2  —  2  p  cos  q  <P  +  p2 1)  =  0        cyclotonic 

(<P  —  a !)•($  —  &  I)2  =  0  simple  shearer 

(0  — al).  (<P  —  b  I)  =  0  special  tonic 

(0  —  a  I)3  =  0  complex  shearer 

(0  —  a  I)2  =  0       special  simple  shearer 

(<P  —  a  I)  =  0  special  tonic. 


CHAPTER  YII 

MISCELLANEOUS  APPLICATIONS 

Quadric  Surfaces 
136.]     If  $  be  any  constant  dyadic  the  equation 

r  .  0.i  =  const.  (1) 

is  quadratic  in  r.  The  constant,  in  case  it  be  not  zero,  may 
be  divided  into  the  dyadic  0  and  hence  the  equation  takes 

the  form 

r  •  0  .  r  =  1,  (1)' 

or  r  •  0  •  r  =  0.  (2) 

The  dyadic  0  may  be  assumed  to  be  self-conjugate.  For  if 
W  is  an  anti-self-conjugate  dyadic,  the  product  r  •  W  •  r  is 
identically  zero  for  all  values  of  r.  The  proof  of  this  state- 
ment is  left  as  an  exercise.  By  Art.  116  any  self-conjugate 
dyadic  is  reducible  to  the  form 

*=±-±y±"         (3) 

-1-          9    •*•     7  <>    •"  •)  \.      / 

a2       bz        c2 
If  r  =  ici  +  yj  +  zk, 

#2      7/2      zz 

r.,.r=±_±|i±_i.  (4) 

Hence  the  equation        r  •  0  •  r  =  1 

represents  a  quadric  surface  real  or  imaginary. 

The  different  cases  which  arise  are  four  in  number.  If  the 
signs  are  all  positive,  the  quadric  is  a  real  ellipsoid.  If  one 
sign  is  negative  it  is  an  hyperboloid  of  one  sheet;  if  two  are 


QUADRIC  SURFACES  373 

negative,  a  hyperboloid  of  two  sheets.  If  the  three  signs  are 
all  negative  the  quadric  is  imaginary.  In  like  manner  the 

equation 

r.0.r  =  0 

is  seen  to  represent  a  cone  which  may  be  either  real  or 
imaginary  according  as  the  signs  are  different  or  all  alike. 

Thus  the  equation 

r  •  0  •  r  =  const. 

represents  a  central  quadric  surface.  The  surface  reduces  to 
a  cone  in  case  the  constant  is  zero.  Conversely  any  central 
quadric  surface  may  be  represented  by  a  suitably  chosen  self- 
conjugate  dyadic  0  in  the  form 

r  •  0  •  r  =  const. 

This  is  evident  from  the  equations  of  the  central  quadric 
surfaces  when  reduced  to  the  normal  form.  They  axe 

x2      y2       zz 
±  —  ±  —  ±  —  =  const. 
a2      62       c2 

i  i      j  j       k  k 

The  corresponding  dyadic  0  is       0  =  ±  — -  ±  —  ±  —%- . 

a2      bz        c2 

The  most  general  scalar  expression  which  is  quadratic  in 
the  vector  r  and  which  consequently  when  set  equal  to  a  con- 
stant represents  a  quadric  surface,  contains  terms  like 

r  •  r,     (r  •  a)  (b  •  )r,     r  •  c,     d  •  c, 

where  a,  b,  c,  d,  e  are  constant  vectors.  The  first  two  terms 
are  of  the  second  order  in  r ;  the  third,  of  the  first  order ;  and 
the  last,  independent  of  r.  Moreover,  it  is  evident  that  these 
four  sorts  of  terms  are  the  only  ones  which  can  occur  in  a 
scalar  expression  which  is  quadratic  in  r. 

But  r  •  r  =  r  •  I  •  r, 

and  (r  •  a)  (b  •  r)  =  r  •  a  b  •  r. 


374  VECTOR  ANALYSIS 

Hence  the  most  general  quadratic  expression  may  be  reduced 

to 

r.0.r  +  r.A  +  (7:=0, 

where  0  is  a  constant  dyadic,  A  a  constant  vector,  and  0 
a  constant  scalar.  The  dyadic  may  be  regarded  as  self- 
conjugate  if  desired. 

To  be  rid  of  the  linear  term  r  •  A,  make  a  change  of  origin 
by  replacing  r  by  r'  —  t. 

(r'  -  t)  -  0  •  (r'  -  t)  +  (r'  -  t)  •  A  +  C  =  0 

r'  •  $  •  r'  -  t  •  0  •  r'  —  r'  •  <P  •  t  +  t  •  0  •  t 

-fr'.A-t.A+C'^O. 

Since  0  is  self-conjugate  the  second  and  third  terms  are 
equal.  Hence 

r'  •  0  •  r'  +  2  r'  •  (|  A  -  4>  •  t)  +  C'  =  0. 

If  now  $  is  complete  the  vector  t  may  be  chosen  so  that 

^A=  0  .  t  or  t  =  l  0-1-  A. 

Hence  the  quadric  is  reducible  to  the  central  form 
r'  •  0  •  r'  =  const. 

In  case  0  is  incomplete  it  is  wm'planar  or  wmlinear  because 
0  is  self  -conjugate.  If  A  lies  in  the  plane  of  0  or  in  the  line 
of  0  as  the  case  may  be  the  equation 


is  soluble  for  t  and  the  reduction  to  central  form  is  still  pos- 
sible. But  unless  A  is  so  situated  the  reduction  is  impossible. 
The  quadric  surface  is  not  a  central  surface. 

The  discussion  and  classification  of  the  various  non-central 
quadrics  is  an  interesting  exercise.  It  will  not  be  taken  up 
here.  The  present  object  is  to  develop  so  much  of  the  theory 


QUADRIC  SURFACES  375 

of  quadric  surfaces  as  will  be  useful  in  applications  to  mathe- 
matical physics  with  especial  reference  to  non-isotropic  media. 
Hereafter  therefore  the  central  quadrics  and  in  particular  the 
ellipsoid  will  be  discussed. 

137.]     The  tangent  plane  may  be  found  by  differentiation. 

r  .  0  .  r  =  1. 

dr  •  0  •  r  +  r  •  0  •  dr  =  0. 
Since  0  is  self-conjugate  these  two  terms  are  equal  and 

dT'0  T  =  0.  (5) 

The  increment  d  r  is  perpendicular  to  0  •  r.  Hence  <P  •  r  is 
normal  to  the  surface  at  the  extremity  of  the  vector  r.  Let 
this  normal  be  denoted  by  N  and  let  the  unit  normal  be  n. 

N  =  0  •  r  (6) 

d>  .  r  0  .  r 


n  = 


r)  •  ($  •  r) 


Let  p  be  the  vector  drawn  from  the  origin  perpendicular  to 
the  tangent  plane,  p  is  parallel  to  n.  The  perpendicular 
distance  from  the  origin  to  the  tangent  plane  is  the  square 
root  of  p  •  p.  It  is  also  equal  to  the  square  root  of  r  •  p. 

r  .  p  =  r  cos  (r,  p)  p  —  p2. 
Hence  r  •  p  =  p  •  p. 

Or  LH  =  r._Z_  =  l. 


p.p         p.p 

But  r.0-r  =  r.H  =  l. 

Hence  inasmuch  as  p  and  N  are  parallel,  they  are  equal. 

0.r  =  N=—  .  (7) 

p.p 


376  VECTOR  ANALYSIS 

On  page  108  it  was  seen  that  the  vector  which  has  the  direc- 
tion of  the  normal  to  a  plane  and  which  is  in  magnitude  equal 
to  the  reciprocal  of  the  distance  from  the  origin  to  the  plane 
may  be  taken  as  the  vector  coordinate  of  that  plane.  Hence 
the  above  equation  shows  that  0  •  r  is  not  merely  normal  to 
the  tangent  plane,  but  is  also  the  coordinate  of  the  plane. 
That  is,  the  length  of  0  •  r  is  the  reciprocal  of  the  distance 
from  the  origin  to  the  plane  tangent  to  the  ellipsoid  at 
the  extremity  of  the  vector  r. 

The  equation  of  the  ellipsoid  in  plane  coordinates  may  be 
found  by  eliminating  r  from  the  two  equations. 

(  r  •  0  •  r  =  1, 


Hence     r  .  d> 

Hence  the  desired  equation  is 

H.^-I-H  =  I.  (8) 

Tf  ^-U^jj4.kk 

*       ~    +        "I  --  ~ 


c2kk. 

Let  r  =  a?i  +  yj  +  zk, 

and  N  =  ui  +  vj  +  w  k, 

where  w,  v,  w  are  the  reciprocals  of  the  intercepts  of  the 
plane  N  upon  the  axes  i,  j,  k.  Then  the  ellipsoid  may  be 
written  in  either  of  the  two  forms  familiar  in  Cartesian 
geometry. 

/y»  A  77  2 


or  N.  Q-i.  N  =  a2  u*  +  bzvz  +  c2wz  =  l.      (10) 


QUADRIC  SURFACES  377 

138.]  The  locus  of  the  middle  points  of  a  system  of 
parallel  chords  in  an  ellipsoid  is  a  plane.  This  plane  is 
called  the  diametral  plane  conjugate  with  the  system  of 
chords.  It  is  parallel  to  the  plane  drawn  tangent  to  the 
ellipsoid  at  the  extremity  of  that  one  of  the  chords  which 
passes  through  the  center. 

Let  r  be  any  radius  vector  in  the  ellipsoid.  Let  s  be  the 
vector  drawn  to  the  middle  point  of  a  chord  parallel  to  a. 

Let  r  =  s  +  x  a. 

If  r  is  a  radius  vector  of  the  ellipsoid 

r  .  0  .  r  =  (s  +  x  a)  •  0  •  (s  +  x  a)  =  1. 
Hence     s  +  0.s-j-2£S.0.a  +  #2a.0.a  =  l. 

Inasmuch  as  the  vector  s  bisects  the  chord  parallel  to  a  the 
two  solutions  of  x  given  by  this  equation  are  equal  in  mag- 
nitude and  opposite  in  sign.  Hence  the  coefficient  of  the 

linear  term  x  vanishes.  A 

s  .  0  •  a  =  0. 

Consequently  the  vector  s  is  perpendicular  to  0  •  a.  The 
locus  of  the  terminus  of  s  is  therefore  a  plane  passed  through 
the  center  of  the  ellipsoid,  perpendicular  to  0  •  a,  and  parallel 
to  the  tangent  plane  at  the  extremity  of  a. 

If  b  is  any  radius  vector  in  the  diametral  plane  conjugate 

with  a, 

b  .  0  •  a  =  0. 

The  symmetry  of  this  equation  shows  that  a  is  a  radius 
vector  in  the  plane  conjugate  with  b.  Let  c  be  a  third  radius 
vector  in  the  ellipsoid  and  let  it  be  chosen  as  the  line  of 
intersection  of  the  diametral  planes  conjugate  respectively 

with  a  and  b.     Then 

a  .  0  .  b  =  0, 

b  .  0  .  c  =  0,  (11) 

c  .  0  .  a  =  0. 


378  VECTOR  ANALYSIS 

The  vectors  a,  b,  c  are  changed  into  0»a,0*b,<P*eby 
the  dyadic  <P.  Let 

a'  =  <P  •  a,       b'  =  0  •  b,       c'  =  <P  •  c. 
The  vectors  a',  b',  c'  form  the  system  reciprocal  to  a,  b,  c. 
For        a  -  a'  =  a  •  0  •  a  =  1,       b  •  b'  =  b  •  4>  •  b  =  1, 

c  .  c'  =  c  •  0  •  c  =  1, 

and         a«b/  =  a«0»b  =  0,     b  •  c'  =  b  •  0  •  c  =  0, 
c  •  a'  =  c  •  0  •  a  =  0. 

The  dyadic  0  may  be  therefore  expressed  in  the  forms 

0  =  a' a'  +  b'V  +  c'c',  (12) 

and  0"1  =  a  a  +  bb  +  cc. 

If  for  convenience  the  three  directions  a,  b,  c,  be  called  a 
system  of  three  conjugate  radii  vectors,  and  if  in  a  similar 
manner  the  three  tangent  planes  at  their  extremities  be  called 
a  system  of  three  conjugate  tangent  planes,  a  number  of 
geometric  theorems  may  be  obtained  from  interpreting  the 
invariants  of  0.  A  system  of  three  conjugate  radii  vectors 
may  be  obtained  in  a  doubly  infinite  number  of  ways. 

The  volume  of  a  parallelepiped  of  which  three  concurrent 
edges  constitute  a  system  of  three  conjugate  radii  vectors  is 
constant  and  equal  in  magnitude  to  the  rectangular  parallele- 
piped constructed  upon  the  three  semi-axes  of  the  ellipsoid. 

For  let  a,  b,  c  be  any  system  of  three  conjugate  axes. 

0-J  =  aa  +  bb  +  cc. 

The  determinant  or  third  of  0~l  is  an  invariant  and  inde- 
pendent of  the  form  in  which  0  is  expressed. 

08-i=[abc]2. 


QUADRIC  SURFACES  379 

But  if  0-1  =  a2  ii  +  Z>2  j  j  +  c2  kk, 

03-i  =  a2  bz  c2. 
Hence  [a  b  c]  =  ale. 

This  demonstrates  the  theorem.  In  like  manner  by  inter- 
preting <P3,  ^V"1,  and  0£  it  is  possible  to  show  that: 

The  sum  of  the  squares  of  the  radii  vectors  drawn  to  an 
ellipsoid  in  a  system  of  three  conjugate  directions  is  constant 
and  equal  to  the  sum  of  the  squares  of  the  semi-axes. 

The  volume  of  the  parallelepiped,  whose  three  concurrent 
edges  are  in  the  directions  of  the  perpendiculars  upon  a  system 
of  three  conjugate  tangent  planes  and  in  magnitude  equal  to 
the  reciprocals  of  the  distances  of  those  planes  from  the 
center  of  the  ellipsoid,  is  constant  and  equal  to  the  reciprocal 
of  the  parallelepiped  constructed  upon  the  semi-axes  of  the 
ellipsoid. 

The  sum  of  the  squares  of  the  reciprocals  of  the  three  per- 
pendiculars dropped  from  the  origin  upon  a  system  of  three 
conjugate  tangent  planes  is  constant  and  equal  to  the  sum  of 
the  squares  of  the  reciprocals  of  the  semi-axes. 

If  i,  j,  k  be  three  mutually  perpendicular  unit  vectors 

0a  =  i  .  <p  .  i  +  j  .  0  .  j  +  k  •  0  •  k, 
4>a-i  =  i.  0-i  .  i  +  j  .  0-i  .  j  +  k  •  0-1  •  k. 

Let  a,  b,  c  be  three  radii  vectors  in  the  ellipsoid  drawn 
respectively  parallel  to  i,  j,  k. 

a.  0  .  a  =  b  •  $  •  b  =  c  •  0  •  c  =  1. 

i  •  0  .  i       j  .  0  .  j      k  .  0  .  k 

Hence         ffia  = \- , — — — -  H 

a  •  <P  •  a     b  •  0-  b       c  •  0»  c 

But  the  three  terms  in  this  expression  are  the  squares  of  the 
reciprocals  of  the  radii  vectors  drawn  respectively  in  the  i,  j, 
k  directions.  Hence : 


N 


380  VECTOR   ANALYSIS 

The  sum  of  the  squares  of  the  reciprocals  of  three  mutually 
perpendicular  radii  vectors  in  an  ellipsoid  is  constant.  And 
in  a  similar  manner:  the  sum  of  the  squares  of  the  perpen- 
diculars dropped  from  the  origin  upon  three  mutually  perpen- 
dicular tangent  planes  is  constant. 

139.]  The  equation  of  the  polar  plane  of  the  point  deter- 
mined by  the  vector  a  is 1 

s  .  0  •  a  =  1.  (13) 

For  let  s  be  the  vector  of  a  point  in  the  polar  plane.     The 
vector  of  any  point  upon  the  line  which  joins  the  terminus  of 

s  and  the  terminus  of  a  is 

ys  +  sa 

x  +  y 
If  this  point  lies  upon  the  surface 

i/  a  _J-    /v»  a  i/  a   _I_    ••>•>  a 

=  1 


x  +  y  x  +  y 

g.0.8+         xy     n.0.i  +  ,    X         a.0.a  =  l. 


O  +  y)2  O  +  2/)2  O  +  2/)2 

If  the  terminus  of  s  lies  in  the  polar  plane  of  a  the  two  values 
of  the  ratio  x  :  y  determined  by  this  equation  must  be  equal 
in  magnitude  and  opposite  in  sign.  Hence  the  term  inxy 
vanishes. 

Hence  s  •  <P  •  a  =  1 

is  the  desired  equation  of  the  polar  plane  of  the  terminus 
of  a. 

Let  a  be  replaced  by  z  a.     The  polar  plane  becomes 


or  s  •  d>  •  a  =  -  • 

z 

1  It  is  evidently  immaterial  whether  the  central  quadric  determined  by  *  be 
real  or  imaginary,  ellipsoid  or  hyperboloid. 


QUADRIC  SURFACES  381 

When  z  increases  the  polar  plane  of  the  terminus  of  z  a 
approaches  the  origin.  In  the  limit  when  z  becomes  infinite 
the  polar  plane  becomes 

s  .  0  •  a  =  0. 

Hence  the  polar  plane  of  the  point  at  infinity  in  the  direction 
a  is  the  same  as  the  diametral  plane  conjugate  with  a.  This 
statement  is  frequently  taken  as  the  definition  of  the  diame- 
tral plane  conjugate  with  a.  In  case  the  vector  a  is  a  radius 
vector  of  the  surface  the  polar  plane  becomes  identical  with 
the  tangent  plane  at  the  terminus  of  a.  The  equation 

s  •  #  •  a  =  1     or  s  •  N  =  1 

therefore  represents  the  tangent  plane. 

The  polar  plane  may  be  obtained  from  another  standpoint 
which  is  important.     If  a  quadric  Q  and  a  plane  P  are  given, 


and  P  =  r  •  c  -  C  =  0, 

the  equation       (r  •  0  •  r  —  1)  +  k  (r  •  c  —  C)2  =  0 

represents  a  quadric  surface  which  passes  through  the  curve 
of  intersection  of  Q  and  P  and  is  tangent  to  Q  along  that 
curve.  In  like  manner  if  two  quadrics  Q  and  Q'  are  given, 


the  equation      (r  •  0  •  r  —  1)  +  k  (r  •  0'  •  r  —  1)  =  0 

represents  a  quadric  surface  which  passes  through  the  curves 
of  intersection  of  Q  and  Q'  and  which  cuts  Q  and  Q'  at  no 
other  points.  In  case  this  equation  is  factorable  into  two 
equations  which  are  linear  in  r,  and  which  consequently  rep- 
resent two  planes,  the  curves  of  intersection  of  Q  and  Q' 
become  plane  and  lie  into  those  two  planes. 


382  VECTOR  ANALYSIS 

If  A  is  any  point  outside  of  the  quadric  and  if  all  the  tangent 
planes  which  pass  through  A  are  drawn,  these  planes  envelop 
a  cone.  This  cone  touches  the  quadric  along  a  plane  curve  — 
the  plane  of  the  curve  being  the  polar  plane  of  the  point  A. 
For  let  a  be  the  vector  drawn  to  the  point  A.  The  equation 
of  any  tangent  plane  to  the  quadric  is 

s  .  0  .  r  =  1. 

If  this  plane  contains  A,  its  equation  is  satisfied  by  a.  Hence 
the  conditions  which  must  be  satisfied  by  r  if  its  tangent 
plane  passes  through  A  are 

a  .  0  .  r  =  1, 
r  •  0  •  r  =  1. 

The  points  r  therefore  lie  in  a  plane  r  •  (0  •  a)  =  1  which 
on  comparison  with  (13)  is  seen  to  be  the  polar  plane  of  A. 
The  quadric  which  passes  through  the  curve  of  intersection 
of  this  polar  plane  with  the  given  quadric  and  which  touches 
the  quadric  along  that  curve  is 

(r  •  0  •  r  -  1)  +  Jfe  (a  •  0  •  r  -  I)2  =  0. 
If  this  passes  through  the  point  A, 

(a  .  d> .  a  -  1)  +  k  (a  .  ffi  •  a  -  I)2  =  0. 
Hence  (r  •  4>  •  r  -  1)  (a  •  0  •  a  -  1)  -  (a  •  0  •  r  -  I)2  =  0. 

By  transforming  the  origin  to  the  point  A  this  is  easily  seen 
to  be  a  cone  whose  vertex  is  at  that  point. 

140.]  Let  0  be  any  self-conjugate  dyadic.  It  is  expres- 
sible in  the  form 

0  =  A  ii  +  ^jj  +  tfkk 

where  A>  B,  C  are  positive  or  negative  scalars.  Further- 
more let 

A  <B  <  0 

0  -  Bl  =  ((7-  J5)  kk  -  (B  -  Ay  ii. 


QUADRIC  SURFACES  383 

Let  V  C  —  B  k  =  c   and    V  B  —  A   i  =  a. 

Then    0 - £I  =  cc-aa=^  j  (c  +  a)(c-a)+(c-a)(c+a)  j. 
Let  c  +  a  =  p     and    c  —  a  =  q. 

Then  0  =  Bl  +  |-(p  q  +  qp).  (14) 

The  dyadic  0  has  been  expressed  as  the  sum  of  a  constant 
multiple  of  the  idemfactor  and  one  half  the  sum 

pq  +  qp. 

The  reduction  has  assumed  tacitly  that  the  constants  -4,  B,  G 
are  different  from  each  other  and  from  zero. 

This   expression  for    0  is  closely  related  to  the  circular 
sections  of  the  quadric  surface 

r  •  0  •  r  =  1. 

Substituting  the  value  of  <P,     r  •  0  •  r  =  1  becomes 

B  r  •  r  +  T  •  p  q  •  r  =  1. 
Let  r  •  p  =  n 

be  any  plane  perpendicular  to  p.     By  substitution 
B  T  •  r  +  n  q  •  r  —  1  =  0. 

This  is  a  sphere  because  the  terms  of  the  second  order  all 
have  the  same  coefficient  B.  If  the  equation  of  this  sphere 
be  subtracted  from  that  of  the  given  quadric,  the  resulting 
equation  is  that  of  a  quadric  which  passes  through  the  inter- 
section of  the  sphere  and  the  given  quadric.  The  difference 

q  •  r  (r  •  p  —  w)  =  0. 

Hence  the  sphere  and  the  quadric  intersect  in  two  plane 
curves  lying  in  the  planes 

q  •  r  =  0     and    r  •  p  =  n. 


384  VECTOR  ANALYSIS 

Inasmuch  as  these  curves  lie  upon  a  sphere  they  are  circles. 
Hence  planes  perpendicular  to  p  cut  the  quadric  in  circles. 
In  like  manner  it  may  be  shown  that  planes  perpendicular  to 
q  cut  the  quadric  hi  circles.  The  proof  may  be  conducted  as 

follows : 

.Z?  r  •  r  +  r  •  p  q  •  r  =  1. 

If  r  is  a  radius  vector  in  the  plane  passed  through  the  center 
of  the  quadric  perpendicular  to  p  or  q,  the  term  r  •  p  q  •  r  van- 
ishes. Hence  the  vector  r  in  this  plane  satisfies  the  equation 

B  r-r  =  l 

and  is  of  constant  length.  The  section  is  therefore  a  circular 
section.  The  radius  of  the  section  is  equal  hi  length  to  the 
mean  semi-axis  of  the  quadric. 

For  convenience  let  the  quadric  be  an  ellipsoid.  The  con- 
stants A,  B,  G  are  then  positive.  The  reciprocal  dyadic  0~l 
may  be  reduced  hi  a  similar  manner. 

i  i      i  i      k  k 

0-1  =  —  +  ti  H 

A       B        C 


Let  f 

Then      0-1  -  —  I  =  f  f  -  dd  =  \\  (f  +  d)  (f  -  d) 

Jo  *   (. 

+  (f-d)(f  +  d)j 

Let  f  +  d  =  u    and    f  -  d  =  v. 

Then  0-i  =  i  I  +  l  (u  v  +  v  u).  (15) 


QUADRIC  SURFACES  385 

The  vectors  u  and  v  are  connected  intimately  with  the  cir- 
cular cylinders  which  envelop  the  ellipsoid 

r  .  0  .  r  =  1     or    N  •  <P~l  •  N  =  1. 
For  -  N  •  N  +  N  •  u  v  •  N  =  1. 

If  now  N  be  perpendicular  to  u  or  v  the  second  term,  namely, 
N  •  u  v  •  N,  vanishes  and  hence  the  equation  becomes 

N  .  N  =  B. 
That  is,  the  vector  N  is  of  constant  length.     But  the  equation 


is  the  equation  of  a  cylinder  of  which  the  elements  and  tan- 
gent planes  are  parallel  to  n.  If  then  N  •  BT  is  constant  the 
cylinder  is  a  circular  cylinder  enveloping  the  ellipsoid.  The 
radius  of  the  cylinder  is  equal  in  length  to  the  mean  semi-axis 
of  the  ellipsoid. 

There  are  consequently  two  planes  passing  through  the 
origin  and  cutting  out  circles  from  the  ellipsoid.  The  normals 
to  these  planes  are  p  and  q.  The  circles  pass  through  the 
extremities  of  the  mean  axis  of  the  ellipsoid.  There  are  also 
two  circular  cylinders  enveloping  the  ellipsoid.  The  direction 
of  the  axes  of  these  cylinders  are  u  and  v.  Two  elements  of 
these  cylinders  pass  through  the  extremities  of  the  mean  axis 
of  the  ellipsoid. 

These  results  can  be  seen  geometrically  as  follows.  Pass 
a  plane  through  the  mean  axis  and  rotate  it  about  that 
axis  from  the  major  to  the  minor  axis.  The  section  is  an 
ellipse.  One  axis  of  this  ellipse  is  the  mean  axis  of  the 
ellipsoid.  This  remains  constant  during  the  rotation.  The 
other  axis  of  the  ellipse  varies  in  length  from  the  major  to  the 
minor  axis  of  the  ellipsoid  and  hence  at  some  stage  must  pass 
through  a  length  equal  to  the  mean  axis.  At  this  stage  of 

25 


386  VECTOR  ANALYSIS 

the  rotation  the  section  is  a  circle.  In  like  manner  consider 
the  projection  or  shadow  of  the  ellipsoid  cast  upon  a  plane 
parallel  to  the  mean  axis  by  a  point  at  an  infinite  distance 
from  that  plane  and  in  a  direction  perpendicular  to  it.  As  the 
ellipsoid  is  rotated  about  its  mean  axis,  from  the  position  in 
which  the  major  axis  is  perpendicular  to  the  plane  of  projec- 
tion to  the  position  hi  which  the  minor  axis  is  perpendicular 
to  that  plane,  the  shadow  and  the  projecting  cylinder  have  the 
mean  axis  of  the  ellipsoid  as  one  axis.  The  other  axis  changes 
from  the  minor  axis  of  the  ellipsoid  to  the  major  and  hence  at 
some  stage  of  the  rotation  it  passes  through  a  value  equal  to 
the  mean  axis.  At  this  stage  the  shadow  and  projecting 
cylinder  are  circular. 

The  necessary  and  sufficient  condition  that  r  be  the  major 
or  minor  semi-axis  of  the  section  of  the  ellipsoid  r  •  d>  •  r  =  1 
by  a  plane  passing  through  the  center  and  perpendicular  to  a 
is  that  a,  r,  and  0  •  r  be  coplanar. 

Let          .  r  •  0  •  r  =  1 

and  r  •  a  =  0. 

Differentiate  :  d  r  •  0  •  r  =  0, 

d  r  .  a  =  0. 
Furthermore  d  r  •  r  =  0, 

if  r  is  to  be  a  major  or  minor  axis  of  the  section ;  for  r  is  a 
maximum  or  a  mininum  and  hence  is  perpendicular  to  dr. 
These  three  equations  show  that  a,  r,  and  0  •  r  are  all  ortho- 
gonal to  the  same  vector  dr.  Hence  they  are  coplanar. 

[a  r    d>  •  r]  =  0.  (16) 

Conversely  if  [a  r    0  •  r]  =  0, 

dr  may  be  chosen    perpendicular   to  their   common  plane. 

Then 

d  r  •  r  =  0. 


QUADRIC  SURFACES  387 

Hence  r  is  a  maximum  or  a  mininum  and  is  one  of  the  prin- 
cipal semi-axes  of  the  section  perpendicular  to  a. 

141.]     It  is  frequently  an  advantage  to  write  the  equation 
of  an  ellipsoid  in  the  form 

r  .  ¥*  .  r  =  1,  (17) 

instead  of  r  •  0  •  r  =  1. 

This  may  be  done  ;  because  if 

i  i     j  j      k  k 

0  —  --  h  —  H  -- 
2        2         2  ' 


is  a  dyadic  such  that  W*  is  equal  to  0.  W  may  be  regarded  as 
a  square  root  of  0  and  written  as  $*.  But  it  must  be  re- 
membered that  there  are  other  square  roots  of  ®  —  for 

example, 

i  i     j  j      kk 

__  i  *  *  _  __ 

a        b         c 

and  —  -  H  -  — 

a        b         c  ' 

For  this  reason  it  is  necessary  to  bear  in  mind  that  the  square 
root  which  is  meant  by  <P*  is  that  particular  one  which  has 
been  denoted  by  ¥. 

The  equation  of  the  ellipsoid  may  be  written  in  the  form 

r  .  W  •  W  •  r  =  1, 
or      .  (?T.r).  (?F.r)  =  l. 

Let  r'  be  the  radius  vector  of  a  unit  sphere.     The  equation  of 

the  sphere  is 

r'  •  r'  =  1. 


388  VECTOR  ANALYSIS 

If  r' '=  ^'rit  becomes  evident  that  an  ellipsoid  may  be 
transformed  into  a  unit  sphere  by  applying  the  operator  ¥ 
to  each  radius  vector  r,  and  vice  versa,  the  unit  sphere  may 
be  transformed  into  an  ellipsoid  by  applying  the  inverse  oper- 
ator W~l  to  each  radius  vector  r'.  Furthermore  if  a,  b,  c  are 
a  system  of  three  conjugate  radii  vectors  in  an  ellipsoid 

a.  ¥2  •  a  =  b  •  2P2  •  b  =  c  •  5F2 
a  .  Wz  •  b  =  b  •  ¥2  .  c  =  c  •  ¥2 

If  for  the  moment  a',  b',  c'  denote  respectively  W  •  a,   W  •  b, 

r.c, 

a' .  a'  =  V  •  b'  =  c' .  c'  =  1, 

a'  •  b'  =  V  •  c'  =  c'  •  a'  =  0. 

Hence  the  three  radii  vectors  a',  b',  c'  of  the  unit  sphere  into 
which  three  conjugate  radii  vectors  in  the  ellipsoid  are  trans- 
formed by  the  operator  ¥~l  are  mutually  orthogonal.  They 
form  a  right-handed  or  left-handed  system  of  three  mutually 
perpendicular  unit  vectors. 

Theorem :  Any  ellipsoid  may  be  transformed  into  any  other 
ellipsoid  by  means  of  a  homogeneous  strain. 

Let  the  equations  of  the  ellipsoids  be 

r  •  0  •  r  =  1, 
and  f  •  W  •  r  =  1. 

By  means  of  the  strain  0*  the  radii  vectors  r  of  the  first 
ellipsoid  are  changed  into  the  radii  vectors  r'  of  a  unit  sphere 

r'=0*T,       r'T'  =  l. 

By  means  of  the  strain  W~*  the  radii  vectors  r'  of  this  unit 
sphere  are  transformed  in  like  manner  into  the  radii  vectors  f 
of  the  second  ellipsoid.  Hence  by  the  product  r  is  changed 
into  r. 

f  =  V-*  .  0i  .  r.  (19) 


QUADRIC  SURFACES  389 

The  transformation  may  be  accomplished  in  more  ways 
than  one.  The  radii  vectors  r'  of  the  unit  sphere  may  be 
transformed  among  themselves  by  means  of  a  rotation  with  or 
without  a  perversion.  Any  three  mutually  orthogonal  unit 
vectors  in  the  sphere  may  be  changed  into  any  three  others. 
Hence  the  semi-axes  of  the  first  ellipsoid  may  be  carried  over 
by  a  suitable  strain  into  the  semi-axes  of  the  second.  The 
strain  is  then  completely  determined  and  the  transformation 
can  be  performed  in  only  one  way. 

142.]  The  equation  of  a  family  of  confocal  quadric  sur- 
faces is 


az  —  n       b'2  —  n       c'2  —  n 

If  r  .  0  •  r  =  1  and  r  •  ¥  •  r  =  1  are  two  surfaces  of  the 

family, 

i  i  j  j  k  k 


i  i  j  j  k  k 

nr  — L_     *  * u 

•*      o  >       •.  o  I 


a2— «,„      62— 7i»      c2  —  n* 

&  &  & 

j:  —   ^tt      —  T2/   ^   1  1  ~f"    Cb      —  72.   ^   1  1   "^~   ( C      ---—  71    )   k  K. 

Hence        0-1  -  y-i=  (wa  -  n^  (ii  +  j  j  +  kk) 

=  (%  -  wi)  T- 
The  necessary  and  sufficient  condition  that  the  two  quadrics 

r  •  0  •  r  =  1 
and  r  •  ¥  •  r  =  1 

be  confocal,  is  that  the  reciprocals  of  0  and  ¥  differ  by  a 
multiple  of  the  idemfactor 

0-i  _  y-i  =  a.i.  (21) 


390  VECTOR  ANALYSIS 

If  two  confocal  quadrics  intersect,  they  do  so  at  right  angles. 
Let  the  quadrics  be        r  •  0  •  r  =  1, 
and  r  •  W  •  r  =  1. 

Let  s  =  0  •  r  and  *'  =  W  .  r, 

r  =  0-1  .  s  and  r  =  W~l  •  s'. 

Then  the  quadrics  may  be  written  in  terms  of  s  and  s'  as 

s  .  0-1  .s  =  l, 
and  s'.  ?T-i.s'  =  l, 

where  by  the  confocal  property, 

0-i  _  yr-i  =  xit 

If  the  quadrics  intersect  at  r  the  condition  for  perpendicularity 
is  that  the  normals  0  •  r  and  W  •  r  be  perpendicular.    That  is, 

s  .  s'  =  0. 
But       r  =  ¥-1 .  s'  =  0-1  •  s  =  (  y~l  +  x  I)  •  s 

=   ¥~l  •  8  +  X  8, 

x  s  -  s'  =  s'  •  y-1  .  s'  -  s  •  5P-1  .  s'=  1  -  s  •  SF-1  .  s'. 
In  like  manner 

r  -  CM  .  s  =  W-*  •  s'  -  (0-1-  aj  I)  •  s'  =  0-1  •  §'  -  a;  s'. 
a?  s  •  s'  =  s  •  0-1  •  s'  —  8  •  0-1  •  s  =  8  •  0-1  •  s'  —  1. 
Add :          2  a;  s  •  s'  =  s  •  (0~J  —  JP"-1)  •  sr  =  a;  s  •  s'. 
Hence  s  •  s'  =  0, 

and  the  theorem  is  proved. 

If  the  parameter  n  be  allowed  to  vary  from  —  oo  to  +  oo  the 
resulting  confocal  quadrics  will  consist  of  three  families  of 
which  one  is  ellipsoids ;  another,  hyperboloids  of  one  sheet  ; 
and  the  third,  hyperboloids  of  two  sheets.  By  the  foregoing 


QUADRIC  SURFACES  391 

theorem  each  surface  of  any  one  family  cuts  every  surface 
of  the  other  two  orthogonally.  The  surfaces  form  a  triply 
orthogonal  system.  The  lines  of  intersection  of  two  families 
(say  the  family  of  one-sheeted  and  the  family  of  two-sheeted 
hyperboloids)  cut  orthogonally  the  other  family  —  the  family 
of  ellipsoids.  The  points  in  which  two  ellipsoids  are  cut  by 
these  lines  are  called  corresponding  points  upon  the  two  ellip- 
soids. It  may  be  shown  that  the  ratios  of  the  components  of 
the  radius  vector  of  a  point  to  the  axes  of  the  ellipsoid 
through  that  point  are  the  same  for  any  two  corresponding 
points. 

For  let  any  ellipsoid  be  given  by  the  dyadic 


The  neighboring  ellipsoid  in  the  family  is  represented  by  the 
dyadic 

W=       U  JJ        ,      kk 

n-»  *^    i  o  •»  *^ 


a?  —  dn      bz  —  dn      c2  —  d  n 
JF-i=  0-1  +  I  dn. 

Inasmuch  as  0  and  ¥  are  homologous  (see  Ex.  8,  p.  330) 
dyadics  they  may  be  treated  as  ordinary  scalars  in  algebra. 
Therefore  if  terms  of  order  higher  than  the  first  in  dn  be 
omitted,  V=*  +  Idn. 

The  two  neighboring  ellipsoids  are  then 

r  .  d>  .  r  =  1, 

and  f  -  (0  +  I  dn)  •  r  =  1. 

By  (19)  r  =  (0  +  I  dn)~*  •  0*  •  r, 

f  =  (I  +  4>  dn)-1*  .  r, 

1  d  n    , 

r  _  (I  _  i  0  dn)  •  r  =  r  -  -~  0  .  r. 


392  VECTOR  ANALYSIS 

The  vectors  f  and  r  differ  by  a  multiple  of  <P  •  r  which  is 
perpendicular  to  the  ellipsoid  (P.  Hence  the  termini  of  f  and 
r  are  corresponding  points,  for  they  lie  upon  one  of  the  lines 
which  cut  the  family  of  ellipsoids  orthogonally.  The  com- 
ponents of  r  and  f  in  the  direction  i  are  r  •  i  =  x  and 

_  m       dn  .  dn  x 

r  .  i  =  a  =  r  •  i  -  —  i.      •  r '-x-  •  -£-  -^  • 

Of  (7  TL 

The  ratio  of  these  components  is          -=!  —  --—• 

x  2  a2 

The  axes  of  the  ellipsoids  in  the  direction  i  are  Va2  —  dn  and 

a.     Their  ratio  is 

, 1  dn  j 

Va2  —  dn      «  —  s  —      -i        dn       x 

— =          2   a   =  I 


2  a2      x 


T    v,  V&2  —  dn      y      j  Vc2  —  dn      z 

In  like  manner =  -  and =  -. 

by  c  z 

Hence  the  ratios  of  the  components  of  the  vectors  r  and  r 
drawn  to  corresponding  points  upon  two  neighboring  ellip- 
soids only  differ  at  most  by  terms  of  the  second  order  in  d  n 
from  the  ratios  of  the  axes  of  those  ellipsoids.  It  follows 
immediately  that  the  ratios  of  the  components  of  the  vectors 
drawn  to  corresponding  points  upon  any  two  ellipsoids,  sepa- 
rated by  a  finite  variation  in  the  parameter  n,  only  differ  at 
most  by  terms  of  the  first  order  in  dn  from  the  ratios  of  the 
axes  of  the  ellipsoids  and  hence  must  be  identical  with  them. 
Thio  completes  the  demonstration. 

The  Propagation  of  Light  in  Crystals1 

143.]  The  electromagnetic  equations  of  the  ether  or  of  any 
infinite  isotropic  medium  which  is  transparent  to  electromag- 
netic waves  may  be  written  in  the  form 

1  The  following  discussion  must  be  regarded  as  mathematical  not  physical. 
To  treat  the  subject  from  the  standpoint  of  physics  would  be  out  of  place  here. 


THE  PROPAGATION  OF  LIGHT  IN  CRYSTALS      393 

/72-n 

Pot—  —  +  UD  +  vF-o,      V.D  =  O      (i) 

ct  t 

where  D  is  the  electric  displacement  satisfying  the  hydro-dy- 
namic equation  V  •  D  =  0,  E  a  constant  of  the  dielectric  meas- 
ured in  electromagnetic  units,  and  V  V  the  electrostatic  force 
due  to  the  function  V.  In  case  the  medium  is  not  isotropic  the 
constant  E  becomes  a  linear  vector  function  (P.  This  function 
is  self-conjugate  as  is  evident  from  physical  considerations. 
For  convenience  it  will  be  taken  as  4  IT  0.  The  equations 
then  become 


VF=0,        V-D=0.    (2) 


Operate  by  V  x  V  x. 


d2  D 

V  x  V  x  Pot  -— r  +47rVxVx0-D  =  0.         (3) 

n    /« 

Ct/    6 

The  last  term  disappears  owing  to  the  fact  that  the  curl  of 
the  derivative  VF"  vanishes  (page  167).  The  equation  may 
also  be  written  as 

d2  D 

Pot  V  X  V  X  -r-y  +47rVxVx0.D  =  0.       (3)' 

But  VxVx=VV«  —  V  •  V. 

Remembering    that    V  •  D  and    consequently   V  •  — —  and 

Ct  v 

d2  D 

V  •  — £  vanish  and  that  Pot  V  •  V  is  equal  to  —  4  TT  the 

Ct  t 

equation  reduces  at  once  to 

?  =  V.V<P.D-VV.0.D,         V  •  D  =  0.         (4) 

Ct  t 

Suppose  that  the  vibration  D  is  harmonic.     Let  r  be  the 
vector  drawn  from  a  fixed  origin  to  any  point  of  space. 


394  VECTOR  ANALYSIS 

Then  D  =  A  cos  (m  •  r  —  n  £) 

where  A  and  m  are  constant  vectors  and  n  a  constant  scalar 
represents  a  train  of  waves.  The  vibrations  take  place  in 
the  direction  A.  That  is,  the  wave  is  plane  polarized.  The 
wave  advances  in  the  direction  m.  The  velocity  v  of  that  ad- 
vance is  the  quotient  of  n  by  m,  the  magnitude  of  the  vector 
m.  If  this  wave  is  an  electromagnetic  wave  in  the  medium 
considered  it  must  satisfy  the  two  equations  of  that  medium. 
Substitute  the  value  of  D  in  those  equations. 

The  value  of  V  •  D,  V  •  V  0  •  D,  and  V  V  •  0  •  D  may  be 
obtained  most  easily  by  assuming  the  direction  i  to  be  coinci- 
dent with  m.  m  •  r  then  reduces  to  m  i  •  r  which  is  equal  to 
m  x.  The  variables  y  and  z  no  longer  occur  in  D.  Hence 

D  =  A  cos  (m  x  —  n  t) 

5D 

V  •  D  =  i  •  -=—  =  —  i  •  A    m  sin  (m  x  —  n  0 
d  X 

V  •  V  0  •  D  =  —  m2  0  .  A    cos  (m  x  —  n  f) 
V  V  •  0  •  D  =  —  m2  i  i  •  0 . A    cos  (mx  —  nf). 
Hence  V  •  D  =  —  m  •  A    sin  (m  •  r  —  n  t~) 

V  .  V  0  •  D  =  —  m  •  m    <P  •  D 

VV.<P.D  =  -mm.<P.D. 

d2D 

Moreover  —  =  —  nz  D. 

d  t* 

Hence  if  the  harmonic  vibration  D  is  to  satisfy  the  equa- 
tions (4)  of  the  medium 

w2  D  =  m  .  m   <P  .  D  —  m  m  •  <P  •  D  (5) 

and  m  •  A  =  0.  (6) 


THE  PROPAGATION  OF  LIGHT  IN  CRYSTALS     395 

The  latter  equation  states  at  once  that  the  vibrations  must 
be  transverse  to  the  direction  m  of  propagation  of  the  waves. 
The  former  equation  may  be  put  in  the  form 


0.D.  (5)' 

Introduce  s  =  -  - 

n 

The  vector  s  is  in  the  direction  of  advance  m.  The  magnitude 
of  s  is  the  quotient  of  m  by  n.  This  is  the  reciprocal  of  the 
velocity  of  the  wave.  The  vector  s  may  therefore  be  called 
the  wave-slowness. 

D  =  s  •  s    0  •  D  —  s  s  •  0  •  D. 
This  may  also  be  written  as 

D  =  -  (s  x  s  x  0  •  D)  =  s  x  (0  •  D)  x  s. 
Dividing  by  the  scalar  factor  cos  (m  x  —  n  £), 

A  =  sx(0.A)xs  =  s»s    0  •  A  —  88.0.  A.     (7) 

It  is  evident  that  the  wave  slowness  s  depends  not  at  all 
upon  the  phrase  of  the  vibration  but  only  upon  its  direction. 
The  motion  of  a  wave  not  plane  polarized  may  be  discussed  by 
decomposing  the  wave  into  waves  which  are  plane  polarized. 

144.]  Let  a  be  a  vector  drawn  in  the  direction  A  of  the 
displacement  and  let  the  magnitude  of  a  be  so  determined 

that  a  -  0  •  a  =  1.  (8) 

The  equation  (7)  then  becomes  reduced  to  the  form 

a  =  sx(0-a)xs  =  s.s   0.a  =  ss.0«a  (9) 

a  •  0  •  a  =  1.  (8) 

These  are  the  equations  by  which  the  discussion  of  the  velocity 
or  rather  the  slowness  of  propagation  of  a  wave  in  different 
directions  in  a  non-isotropic  medium  may  be  carried  on. 

a  •  a  =  s  •  s     a  •  0  •  a  =  s  •  s.  (10} 


396  VECTOR  ANALYSIS 

Hence  the  wave  slowness  a  due  to  a  displacement  in  the 
direction  a  is  equal  in  magnitude  (but  not  in  direction)  to  the 
radius  vector  drawn  in  the  ellipsoid  a  •  0  •  a  =  1  in  that 
direction. 

axa  =  0  =  s«s    a  x  0  •  a  —  a  x  s    a  •  0  •  a 
0  =  s.s(ax0»a)'0«a  =  axs.<P«a    s  •  0  •  a. 
But  the  first  term  contains  0  •  a  twice  and  vanishes.     Hence 
a  x  s  •  0  •  a  =  [a    s     0  .  a]  =  0.  (11) 

The  wave-slowness  s  therefore  lies  in  a  plane  with  the 
direction  a  of  displacement  and  the  normal  0  •  a  drawn  to  the 
ellipsoid  a  •  0  •  a  =  1  at  the  terminus  of  a.  Since  s  is  perpen- 
dicular to  a  and  equal  in  magnitude  to  a  it  is  evidently  com- 
pletely determined  except  as  regards  sign  when  the  direction 
a  is  known.  Given  the  direction  of  displacement  the  line  of 
advance  of  the  wave  compatible  with  the  displacement  is  com- 
pletely determined,  the  velocity  of  the  advance  is  likewise 
known.  The  wave  however  may  advance  in  either  direction 
along  that  line.  By  reference  to  page  386,  equation  (11)  is  seen 
to  be  the  condition  that  a  shall  be  one  of  the  principal  axes  of 
the  ellipsoid  formed  by  passing  a  plane  through  the  ellipsoid 
perpendicular  to  s.  Hence  for  any  given  direction  of  advance 
there  are  two  possible  lines  of  displacement.  These  are  the 
principal  axes  of  the  ellipse  cut  from  the  ellipsoid  a  •  0  •  a  =  1 
by  a  plane  passed  through  the  center  perpendicular  to  the 
line  of  advance.  To  these  statements  concerning  the  deter- 
minateness  of  s  when  a  is  given  and  of  a  when  s  is  given  just 
such  exceptions  occur  as  are  obvious  geometrically.  If  a  and 
<P  •  a  are  parallel  s  may  have  any  direction  perpendicular  to  a. 
This  happens  when  a  is  directed  along  one  of  the  principal 
axes  of  the  ellipsoid.  If  s  is  perpendicular  to  one  of  the 
circular  sections  of  the  ellipsoid  a  may  have  any  direction  in  the 
plane  of  the  section. 


THE  PROPAGATION  OF  LIGHT  IN  CRYSTALS      397 

When  the  direction  of  displacement  is  allowed  to  vary  the 
slowness  a  varies.  To  obtain  the  locus  of  the  terminus  of  s,  a 
must  be  eliminated  from  the  equation 

a  =  s  •  s    0  •  &  —  ss  •  0  •  a 
or  (I  -  s  •  s    <P  +  s  s  •  <P)  •  a  =  0.  (12) 

The  dyadic  in  the  parenthesis  is  planar  because  it  annihilates 
vectors  parallel  to  a.  The  third  or  determinant  is  zero.  This 
gives  immediately 

(I—  s.s<P  +  ss.<P)3  =  0, 
or  (tf-i-s.  s   I  +  ss)3  =  0.  (13) 

This  is  a  scalar  equation  in  the  vector  s.  It  is  the  locus  of 
the  extremity  of  s  when  a  is  given  all  possible  directions.  A 
number  of  transformations  may  be  made.  By  Ex.  19,  p.  331, 

(  0  +  e  f  )3  =  0B  +  e  •  02  •  f  =  03  +  e  •  0^1  •  f  <P3. 
Hence 


Dividing  out  the  common  factor  and  remembering  that  0  is 
self-conjugate. 

1  +  s.  (0-l-&»  si)-1*  s  =  0. 


S.  1.8  0 

\-  8  •  •  S  =  0 

8-8  1-8.  S  0 

S  •  S  0 


8  * 


Hence  B  •  -  -  -=  •  s  =  0.  (14) 

I  —  s  •  s  0 


Let 


398  VECTOR  ANALYSIS 


i-.         \-- 

?/    \    &2/    I    c2 


Let          s  =  ic  i  +  y  j  +  2  k    and    s2  =  x2  +  y2  +  z2. 
Then  the  equation  of  the  surface  in  Cartesian  coordinates  is 


,  , 

T  "         ~~cf  T 


l-fl 


(14)' 


The   equation  in  Cartesian  coordinates  may  be  obtained 

directly  from 

(0-1  _8.  s  1  +  ss)3:=0. 


The  determinant  of  this  dyadic  is 

a2  —  s2  +  xz          x  y  x  z 

x  y  Z>2  —  s2  +  y2         y  z 

x  z  y  z  cz  —  s2  + 


=  0.        (13)' 


By  means  of  the  relation  s2  =  xz  +  y2  +  z2  this  assumes  the 

forms 

x2  z  z 


_ 


a2  ic2          Z>2  yz          c2  z2 
I *      _i ==  0 

9  o     I         9  •,  9  T       9  9  **i 

s'5  —  a*      sz  —  o'2      s'5  —  c^ 


= 


This  equation  appears  to  be  of  the  sixth  degree.  It  is  how- 
ever of  only  the  fourth.  The  terms  of  the  sixth  order  cancel 
out. 

The  vector  s  represents  the  wave-slowness.     Suppose  that  a 
plane  wave  polarized  in  the  direction  a  passes  the  origin  at  a 


THE  PROPAGATION  OF  LIGHT  IN  CRYSTALS      399 

certain  instant  of  time  with  this  slowness.  At  the  end  of  a 
unit  of  time  it  will  have  travelled  in  the  direction  s,  a  distance 
equal  to  the  reciprocal  of  the  magnitude  of  s.  The  plane  will 
be  in  this  position  represented  by  the  vector  s  (page  108). 

If  *  =  ui  +  vj  +  wls. 

the  plane  at  the  expiration  of  the  unit  tune  cuts  off  intercepts 
upon  the  axes  equal  to  the  reciprocals  of  u,  v,  w.  These 
quantities  are  therefore  the  plane  coordinates  of  the  plane. 
They  are  connected  with  the  coordinates  of  the  points  in  the 
plane  by  the  relation 

If  different  plane  waves  polarized  in  all  possible  different 
directions  a  be  supposed  to  pass  through  the  origin  at  the 
same  instant  they  will  envelop  a  surface  at  the  end  of  a  unit 
of  time.  This  surface  is  known  as  the  wave-surface.  The 
perpendicular  upon  a  tangent  plane  of  the  wave-surface  is  the 
reciprocal  of  the  slowness  and  gives  the  velocity  with  which 
the  wave  travels  in  that  direction.  The  equation  of  the  wave- 
surface  in  plane  coordinates  u,  v,  w  is  identical  with  the  equa- 
tion for  the  locus  of  the  terminus  of  the  slowness  vector  B. 
The  equation  is 

uz  v2  w^ 

P  ^  ^  •*  72  =  °  (15) 

^o^o^o  \/ 

9  ~T""*>  Q 

a4  o*  c 

where  s2  =  u2  +  vz  +  w*.  This  may  be  written  in  any  of  the 
forms  given  previously.  The  surface  is  known  as  FresneVs 
Wave-Surface.  The  equations  in  vector  form  are  given  on 
page  397  if  the  variable  vector  s  be  regarded  as  determining  a 
plane  instead  of  a  point. 

145.]  In  an  isotropic  medium  the  direction  of  a  ray  of 
light  is  perpendicular  to  the  wave-front.  It  is  the  same  as 
the  direction  of  the  wave's  advance.  The  velocity  of  the  ray 


400  VECTOR  ANALYSIS 

is  equal  to  the  velocity  of  the  wave.  In  a  non-isotropic 
medium  this  is  no  longer  true.  The  ray  does  not  travel  per- 
pendicular to  the  wave-front  —  that  is,  in  the  direction  of  the 
wave's  advance.  And  the  velocity  with  which  the  ray  travels 
is  greater  than  the  velocity  of  the  wave.  In  fact,  whereas  the 
wave-front  travels  off  always  tangent  to  the  wave-surface,  the 
ray  travels  along  the  radius  vector  drawn  to  the  point  of  tan- 
gency  of  the  wave-plane.  The  wave-planes  envelop  the 
wave-surface;  the  termini  of  the  rays  are  situated  upon  it. 
Thus  in  the  wave-surface  the  radius  vector  represents  in  mag- 
nitude and  direction  the  velocity  of  a  ray  and  the  perpen- 
dicular upon  the  tangent  plane  represents  in  magnitude  and 
direction  the  velocity  of  the  wave.  If  instead  of  the  wave- 
surface  the  surface  which  is  the  locus  of  the  extremity  of  the 
wave  slowness  be  considered  it  is  seen  that  the  radius  vector 
represents  the  slowness  of  the  wave;  and  the  perpendicular 
upon  the  tangent  plane,  the  slowness  of  the  ray. 

Let  v'  be  the  velocity  of  the  ray.  Then  s  •  v'  =  1  because 
the  extremity  of  v'  lies  in  the  plane  denoted  by  s.  Moreover 
the  condition  that  v'  be  the  point  of  tangency  gives  d  v'  per- 
pendicular to  s.  In  like  manner  if  s'  be  the  slowness  of  the 
ray  and  v  the  velocity  of  the  wave,  s'  •  v  =  1  and  the  condition 
of  tangency  gives  d  &'  perpendicular  to  v.  Hence 

s  •  v'  =  1  and  s'  •  v  =  1,  (16) 

and    s  •  d  v'  =  0,  v  •  d  s'  -  0,  v'  •  d  s  =  0,   s'  -  d  v  =  0, 
v'  may  be  expressed  in  terms  of  a,  s,  and  0  as  follows. 

a  =  g  .  g  0  .  a  _  s  s  •  0  •  a, 
d&  =  2a  •  ds  <p  .  a_s  •  <P  •  ads  +  s  •  s  </>  •  da 
—  s  da  •  0  •  a  =  s  s  •  d>  •  da.. 

Multiply  by  a  and  take  account  of  the  relations  a  •  s  =  0  and 
a  .  0  .  d  a  =  0  and  a  •  a  =  s  •  s.     Then 


THE  PROPAGATION  OF  LIGHT  IN  CRYSTALS      401 

B  •  ds  —  a.  •  ds  s  •  <P  •  a  =  0, 
or  d  s  •  (s  —  a  s  •  0  •  a)  =  0. 

But  since  v'  •  d  s  =  0,  v'  and  s  —  a  s  •  0  •  a  have  the  same 
direction. 

v'  =  x  (s  —  a  s  •  0  •  a), 

s  •  v'  =  #  (s  •  s  —  s  •  a  s  •  CP  •  a)  =  a;  s  •  s. 

s  —  a  s  .  0  .  a 

Hence  v'  = ,  (17) 

s  •  s 

s  .  0 • a  —  a •  $«as«0»a 

v' .  ^  .  a  = =  0. 

s  •  s 

Hence  the  ray  velocity  v'  is  perpendicular  to  0  •  a,  that  is,  the 
ray  velocity  lies  in  the  tangent  plane  to  the  ellipsoid  at  the 
extremity  of  the  radius  vector  a  drawn  in  the  direction  of  the 
displacement.  Equation  (17)  shows  that  v'  is  coplanar  with 
a  and  s.  The  vectors  a,  s,  0  •  a,  and  v'  therefore  He  in  one 
plane.  In  that  plane  s  is  perpendicular  to  a ;  and  v',  to  0  •  a. 
The  angle  from  s  to  v'  is  equal  to  the  angle  from  a  to  0  •  a. 

Making  use  of   the   relations   already  found  (8)  (9)  (11) 
(16)  (17),  it  is  easy  to  show  that  the  two  systems  of  vectors 

a,     v',     a  x  v'     and     0  •  a,     s,     (0  •  a)  X  s 

are  reciprocal  systems.  If  0  •  a  be  replaced  by  a'  the  equa- 
tions take  on  the  symmetrical  form 

s  .  a  —  0  s  •  s  =  a  •  a  a  •  a'  =  1, 

v'.a'  =  0          v'-v'  =  a'.a'          s  -  v' =  1, 

a  =  s  x  a'  x  s  a'  =  v'  x  a  x  v'         (18) 

s  =  a  x  v'  x  a  v'  =  a'  x  s  x  a' 

a  .  0  •  a  =  1  a'  •  0-1  •  a'  =  1. 

Thus  a  dual  relation  exists  between  the  direction  of  displace- 
ment, the  ray- velocity,  and  the  ellipsoid  0  on  the  one  hand ; 

26 


402  VECTOR   ANALYSIS 

and  the  normal  to  the  ellipsoid,  the  wave-slowness,  and  the 
ellipsoid  0  ~l  on  the  other. 

146.]  It  was  seen  that  if  s  was  normal  to  one  of  the  cir- 
cular sections  of  0  the  displacement  a  could  take  place  in  any 
direction  in  the  plane  of  that  section.  For  all  directions  in 
this  plane  the  wave-slowness  had  the  same  direction  and  the 
same  magnitude.  Hence  the  wave-surface  has  a  singular 
plane  perpendicular  to  s.  This  plane  is  tangent  to  the  surface 
along  a  curve  instead  of  at  a  single  point.  Hence  if  a  wave 
travels  in  the  direction  s  the  ray  travels  along  the  elements  of 
the  cone  drawn  from  the  center  of  the  wave-surface  to  this 
curve  in  which  the  singular  plane  touches  the  surface.  The 
two  directions  s  which  are  normal  to  the  circular  sections  of  0 
are  called  the  primary  optic  axes.  These  are  the  axes  of  equal 
wave  velocities  but  unequal  ray  velocities. 

In  like  manner  v'  being  coplanar  with  a  and  0  •  a 

[<P  .  a  v'  a]  =  [a'  v'  0~l  •  a']  =  0. 

The  last  equation  states  that  if  a  plane  be  passed  through 
the  center  of  the  ellipsoid  @~l  perpendicular  to  v',  then  a' 
which  is  equal  to  0  •  a  will  be  directed  along  one  of  the  prin- 
cipal axes  of  the  section.  Hence  if  a  ray  is  to  take  a  definite 
direction  a'  may  have  one  of  two  directions.  It  is  more  con- 
venient however  to  regard  v'  as  a  vector  determining  a  plane. 
The  first  equation 

[0  .  a  v'  a]  ==  0 

states  that  a  is  the  radius  vector  drawn  in  the  ellipsoid  0  to 
the  point  of  tangency  of  one  of  the  principal  elements  of  the 
cylinder  circumscribed  about  0  parallel  to  v' :  if  by  a  principal 
element  is  meant  an  element  passing  through  the  extremities 
of  the  major  or  minor  axes  of  orthogonal  plane  sections 
of  that  cylinder.  Hence  given  the  direction  v'  of  the  ray,  the 
two  possible  directions  of  displacement  are  those  radii  vectors 


VARIABLE  DYADICS  403 

of  the  ellipsoid  which  lie  in  the  principal  planes  of  the  cylin- 
der circumscribed  about  the  ellipsoid  parallel  to  v'. 

If  the  cylinder  is  one  of  the  two  circular  cylinders  which 
may  be  circumscribed  about  0  the  direction  of  displacement 
may  be  any  direction  in  the  plane  passed  through  the  center 
of  the  ellipsoid  and  containing  the  common  curve  of  tangency 
of  the  cylinder  with  the  ellipsoid.  The  ray-velocity  for  all 
these  directions  of  displacement  has  the  same  direction  and 
the  same  magnitude.  It  is  therefore  a  line  drawn  to  one 
of  the  singular  points  of  the  wave-surface.  At  this  singukr 
point  there  are  an  infinite  number  of  tangent  planes  envelop- 
ing a  cone.  The  wave-velocity  may  be  equal  in  magnitude 
and  direction  to  the  perpendicular  drawn  from  the  origin  to 
any  of  these  planes.  The  directions  of  the  axes  of  the  two 
circular  cylinders  circumscriptible  about  the  ellipsoid  0  are 
the  directions  of  equal  ray-velocity  but  unequal  wave-velocity. 
They  are  the  radii  drawn  to  the  singular  points  of  the  wave- 
surface  and  are  called  the  secondary  optic  axes.  If  a  ray 
travels  along  one  of  the  secondary  optic  axes  the  wave  planes 
travel  along  the  elements  of  a  cone. 

Variable  Dyadics.     The  Differential  and  Integral  Calculus 

147.]  Hitherto  the  dyadics  considered  have  been  constant. 
The  vectors  which  entered  into  their  make  up  and  the  scalar 
coefficients  which  occurred  in  the  expansion  in  nonion  form 
have  been  constants.  For  the  elements  of  the  theory  and  for 
elementary  applications  these  constant  dyadics  suffice.  The 
introduction  of  variable  dyadics,  however,  leads  to  a  simplifica- 
tion and  unification  of  the  differential  and  integral  calculus  of 
vectors,  and  furthermore  variable  dyadics  become  a  necessity 
in  the  more  advanced  applications  —  for  instance,  in  the  theory 
of  the  curvature  of  surfaces  and  in  the  dynamics  of  a  rigid 
body  one  point  of  which  4s  fixed. 


404  VECTOR  ANALYSIS 

Let  W  be  a  vector  function  of  position  in  space.     Let  r  be 
the  vector  drawn  from  a  fixed  origin  to  any  point  in  space. 


dr  =  dxi  +  dy  j  +  dz  k, 

9W  ,         9W  9W 

d  W  =  d  x  ^—  +  a  y  -  --  h  d  z  -7—. 
ax  v  y  <y  z 

(    9W        9W         9W) 
Hence        d  W  =  d  r  •  \i  -  --  i-  j  -  --  h  k  —  -  >  . 

(     dx  d  y  dz  ) 

The  expression  enclosed  in  the  braces  is  a  dyadic.  It  thus 
appears  that  the  diif  erential  W  is  a  linear  function  of  d  r,  the 
differential  change  of  position.  The  antecedents  are  i,  j,  k, 
and  the  consequents  the  first  partial  derivatives  of  W  with  re- 
spect of  #,  2/,  z.  The  expression  is  found  in  a  manner  precisely 
analogous  to  del  and  will  in  fact  be  denoted  by  V  W. 

(1) 

Then  <2W  =  dr.VW.  (2) 

This  equation  is  like  the  one  for  the  differential  of  a  scalar 

function  F. 

dV=dr  •  VF. 

It  may  be  regarded  as  defining  VW.  If  expanded  into 
nonion  form  VW  becomes 


dx  Qx  Qx 

9Y 


3y 


9Y  9Z 

+     iT-+kj^-+kk^r-, 
oz  dz  a  z 

if  W 


VARIABLE  DYADICS  405 

The  operators  V  •  and  V  x  which  were  applied  to  a  vector 
function  now  become  superfluous  from  a  purely  analytic 
standpoint.  For  they  are  nothing  more  nor  less  than  the 
scalar  and  the  vector  of  the  dyadic  V  W. 


divW  =  V.  W  =  (VW)^  (4) 

curl  W  =  V  x  W  =  (V  W)x.  (5) 

The  analytic  advantages  of  the  introduction  of  the  variable 
dyadic  VW  are  therefore  these.  In  the  first  place  the  oper- 
ator V  may  be  applied  to  a  vector  function  just  as  to  a  scalar 
function.  In  the  second  place  the  two  operators  V  •  and  V  x 
are  reduced  to  positions  as  functions  of  the  dyadic  V  W.  On 
the  other  hand  from  the  standpoint  of  physics  nothing  is  to 
be  gained  and  indeed  much  may  be  lost  if  the  important  in- 
terpretations of  V  •  W  and  V  x  W  as  the  divergence  and  curl 
of  W  be  forgotten  and  their  places  taken  by  the  analytic  idea 
of  the  scalar  and  vector  of  VW. 

If  the  vector  function  W  be  the  derivative  of  a  scalar 
function  V, 

=dT'  VVT, 


32  y  Qzy  Qzy 

where       VVF=  ii  ^r  +  ij  =—  ^-  +  ik  7:—=-, 
d  x*  d  x  dy  d  x  dz 


, 
J  J  +  J  k 


o 
d  y  3  x  dy*  dy 


,  v.  V 

~f~  fcj  o      n        *    ^J  n      o        '     *        n       ' 

d  z  3  x  dz  0  y  o  z* 

The  result  of  applying  V  twice  to  a  scalar  function  is  seen  to 
be  a  dyadic.  This  dyadic  is  self  -conjugate.  Its  vector  V  X  V  V 
is  zero  ;  its  scalar  V  •  V  V  is  evidently 

52  Y     92V     52F 

V.V  V=  (VV  V)a  =  -=-j  +  .-s-j  +  -^-T. 
5  xz      d2      3  zz 


406  •       VECTOR  ANALYSIS 

If  an  attempt  were  made  to  apply  the  operator  V  symboli- 
cally to  a  scalar  function  V  three  times,  the  result  would  be  a 
sum  of  twenty-seven  terms  like 

9s  V  93T 

1  1  1  «—  r,    i  j  k  *  —  =  —  ^-  ,  etc. 

d  x*  a  x  d  y  dz 

This  is  a  triadic.  Three  vectors  are  placed  in  juxtaposition 
without  any  sign  of  multiplication.  Such  expressions  will 
not  be  discussed  here.  In  a  similar  manner  if  the  operator  V 
be  applied  twice  to  a  vector  function,  or  once  to  a  dyadic  func- 
tion of  position  in  space,  the  result  will  be  a  triadic  and  hence 
outside  the  limits  set  to  the  discussion  here.  The  operators 
V  x  and  V  •  may  however  be  applied  to  a  dyadic  0  to  yield 
respectively  a  dyadic  and  a  vector. 

90  90  90 

Vx0  =  lx-5-r  +  jx3-  +  kx-r-,        (7) 

ax  d  y  3  z 

90          90  90 

V  .  0  =  i  •  ^-  +  j  •  -r-  +  k  .  -r-.  (8) 

3  x  dy  3  z 

If  V  =  ui  +  vj  +  wk, 

where  n,  v,  w  are  vector  functions  of  position  in  space, 

Vx  $  =  V  x  u  i  +  Vxvj  +  Vx  wk,       (7)' 
and  V  •  <P  =  V  •  u  i  +  V  •  v  j  +  V  •  w  k.  (8)' 

Or  if  0  =  i  u  +  j  v  +  k  w, 


- 
9y 

and  V.0  =  ^  +  ^  +  ^.  (8)" 

d  x      d  y       dz 

In  a  similar  manner  the  scalar  operators  (a  •  V)  and  (V  •  V) 
may  be  applied  to  0.     The  result  is  in  each  case  a  dyadic, 


IE  DYADIC  S 

407 

90 

9  x 

90 

^2  ~n         ' 

9y 

90 

9  z  ' 

(9) 

9*0 

92  0      i 

J_                   _L 

1*0 

nm 

(V  .  V)  0  =  „ 

9  x2       9  ?/2       9  z2 

The  operators  a  •  V  and  V  •  V  as  applied  to  vector  func- 
tions are  no  longer  necessarily  to  be  regarded  as  single  oper- 
ators. The  individual  steps  may  be  carried  out  by  means  of 
the  dyadic  VW. 

(a  •  V)  W  =  a  •  (V  W)  =  a  •  V  W, 
(V  .  V)  W  =  V  •  (V  W)  =  V  .  V  W. 

But  when  applied  to  a  dyadic  the  operators  cannot  be  inter- 
preted as  made  up  of  two  successive  steps  without  making  use 
of  the  triadic  V  0.  The  parentheses  however  may  be  removed 
without  danger  of  confusion  just  as  they  were  removed  in 
case  of  a  vector  function  before  the  introduction  of  the  dyadic. 
Formulas  similar  to  those  upon  page  176  may  be  given  for 
differentiating  products  in  the  case  that  the  differentiation 
lead  to  dyadics. 

V  (u  v)  =  V  u  v  +  u  V  v, 

V(vxw)=Vvxw  —  Vwxv, 

Vx  (v  x  w)  ==  w  •  V  v  —  V  •  v  w  —  v  •  V  w  +  V  •  w  v, 

V  (v  •  w)  —  V  v  •  w  +  V  w  •  v, 

V  •  (v  w)  =  V  •  v  w  +  v  •  V  w. 

Vx  (v  w)  =  V  x  v  w  —  vxVw, 

V  •  (u  0~)  =  V  u  •  0  +  u  V  •  0, 

VxVx  0  =  V  V  •  0  —  V  .  V  </>,  etc. 

The  principle  in  these  and  all  similar  cases  is  that   enun- 
ciated before,  namely :     The  operator  V  may  be  treated  sym- 


408  VECTOR  ANALYSIS 

bolically  as  a  vector.  The  differentiations  which  it  implies 
must  be  carried  out  in  turn  upon  each  factor  of  a  product 
to  which  it  is  applied.  Thus 

V  x  (vw)  =  [V  x  (vw)]v  +  [V  x  (vw)]^ 

[V  x  (v  w)]w  =  V  x  v  w, 
[V  x  (v  w)]v  =  —  [v  x  V  w]v  =  —  v  x  V  w. 
Hence  Vx  (vw)  =  Vxvw  —  vxVw. 

Again    V  (v  X  w)  =  [V  (v  x  w)]v  +  [V  (v  x  w)]^ 

[V  (v  x  w)]w  =  V  v  x  w, 
[V  (v  x  w)]v  =  [—  v  (w  x  v)]v  =  —  V  w  x  v. 
Hence  V(vxw)  =  Vvxw  —  v^xv. 

148.]  It  was  seen  (Art.  79)  that  if  C  denote  a  curve  of 
which  the  initial  point  is  r0  and  the  final  point  is  r  the  line  in- 
tegral of  the  derivative  of  a  scalar  function  taken  along  the 
curve  is  equal  to  the  difference  between  the  values  of  that 
function  at  r  and  r0. 

f«Zr.VF=F(r)-F(r0). 

*>  c 

In  like  manner    idr.VW  —  W(r)—  W  (r0), 
J  c 

and  id  r  •  V  W  =  0. 

«/o 

It  may  be  well  to  note  that  the  integrals 

fdr.VW     and       fvw  •  dr 

are  by  no  means  the  same  thing.  VW  is  a  dyadic.  The 
vector  d  r  cannot  be  placed  arbitrarily  upon  either  side  of  it. 


VARIABLE   DYADICS  409 

Owing  to  the  fundamental  equation  (2)  the  differential  dr 
necessarily  precedes  VW.  The  differentials  must  be  written 
before  the  integrands  in  most  cases.  For  the  sake  of  uni- 
formity they  always  will  be  so  placed. 

Passing  to  surface  integrals,  the  following  formulae,  some 
of  which  have  been  given  before  and  some  of  which  are  new, 
may  be  mentioned. 

f  f  dax  VF=   t  dr  V 

Cfd&x  VW=Cdi  W 

ffda»VxW  =  idi'W 

if   d&  •  V  x  <P  =   /  dr  •  0. 

The  line  integrals  are  taken  over  the  complete  bounding  curve 
of  the  surface  over  which  the  surface  integrals  are  taken.  In 
like  manner  the  following  relations  exist  between  volume  and 
surface  integrals. 


fffd,  V.W 

f  f  fdv  V  X  W=   C  Cda,  X  W 

///«••  y  •*-//«•••• 

f  f  f^V    VX    0=:CCdSLX    $. 


410  VECTOR  ANALYSIS 

The  surface  integrals  are  taken  over  the  complete  bounding 
surface  of  the  region  throughout  which  the  volume  integrals 
are  taken. 

Numerous  formulae  of  integration  by  parts  like  those  upon 
page  250  might  be  added.  The  reader  will  find  no  difficulty  in 
obtaining  them  for  himself.  The  integrating  operators  may 
also  be  extended  to  other  cases.  To  the  potentials  of  scalar 
and  vector  functions  the  potential,  Pot  <P,  of  a  dyadic  may  be 
added.  The  Newtonian  of  a  vector  function  and  the  Lapla- 
cian  and  Maxwellian  of  dyadics  may  be  denned. 


Pot 


'-/// 


New  W  =  fff  '"  w  <>" 


Lap  *  =  fff  r 


*    12 

The  analytic  theory  of  these  integrals  may  be  developed  as 
before.  The  most  natural  way  in  which  the  demonstrations 
may  be  given  is  by  considering  the  vector  function  W  as  the 
sum  of  its  components, 

Yj  +  Zk 


and  the  dyadic  0  as  expressed  with  the  constant  consequents 
i,  j,  k  and  variable  antecedents  n,  v,  w,  or  vice  versa, 


These  matters  will  be  left  at  this  point.  The  object  of  en- 
tering upon  them  at  all  was  to  indicate  the  natural  extensions 
which  occur  when  variable  dyadics  are  considered.  These  ex- 
tensions differ  so  slightly  from  the  simple  cases  which  have 


THE   CURVATURE   OF  SURFACES  411 

gone  before  that  it  is  far  better  to  leave  the  details  to  be  worked 
out  or  assumed  from  analogy  whenever  they  may  be  needed 
rather  than  to  attempt  to  develop  them  in  advance.  It  is  suffi- 
cient merely  to  mention  what  the  extensions  are  and  how  they 
may  be  treated. 

The  Curvature  of  Surfaces1 

149.  ]  There  are  two  different  methods  of  treating  the  cur- 
vature of  surfaces.  In  one  the  surface  is  expressed  in  para- 
metic  form  by  three  equations 

x  =fi  <X  *0»     y  =/a  O'  *0»     z  =/8  (w>  v)» 
or  r  =  f  (u,  t>). 

This  is  analogous  to  the  method  followed  (Art.  57)  in  dealing 
with  curvature  and  torsion  of  curves  and  it  is  the  method 
employed  by  Fehr  in  the  book  to  which  reference  was  made. 
In  the  second  method  the  surface  is  expressed  by  a  single 
equation  connecting  the  variables  x,y,z  —  thus 


,  z)  =  0. 

The  latter  method  of  treatments  affords  a  simple  application  of 
the  differential  calculus  of  variable  dyadics.  Moreover,  the 
dyadics  lead  naturally  to  the  most  important  results  connected 
with  the  elementary  theory  of  surfaces. 

Let  r  be  a  radius  vector  drawn  from  an  arbitrary  fixed 
origin  to  a  variable  point  of  the  surface.  The  increment  d  r 
lies  in  the  surface  or  in  the  tangent  plane  drawn  to  the  surface 
at  the  terminus  of  r. 

dF=dr*  VF=Q. 

Hence  the  derivative  V.Fis  collinear  with  the  normal  to  the 
surface.  Moreover,  inasmuch  as  F  and  the  negative  of  F  when 

1  Much  of  what  follows  is  practically  free  from  the  use  of  dyadics.  This  is 
especially  true  of  the  treatment  of  geodetics,  Arts.  155-157. 


412  VECTOR  ANALYSIS 

equated  to  zero  give  the  same  geometric  surface,  V  F  may  be 
considered  as  the  normal  upon  either  side  of  the  surface.  In 
case  the  surface  belongs  to  the  family  defined  by 

F  (#,  y,  z)  =  const. 

the  normal  V  F  lies  upon  that  side  upon  which  the  constant 
increases.  Let  V  F  be  represented  by  N  the  magnitude  of 
which  may  be  denoted  by  N,  and  let  n  be  a  unit  normal  drawn 
in  the  direction  of  N.  Then 


(1) 


If  &  is  the  vector  drawn  to  any  point  in  the  tangent  plane  at 
the  terminus  of  r,  s—  r  and  n  are  perpendicular.  Consequently 
the  equation  of  the  tangent  plane  is 

(s-r).  V^=0. 
and  in  like  manner  the  equation  of  the  normal  line  is 

(s-r)  X  V^=0, 
or  s  =  r  +  kV  F 

where  k  is  a  variable  parameter.     These  equations  may  be 
translated  into  Cartesian  form  and  give  the  familiar  results. 

150.]  The  variation  dn  of  the  unit  normal  to  a  surface 
plays  an  important  part  in  the  theory  of  curvature,  dn.  is 
perpendicular  to  n  because  n  is  a  unit  vector. 

n  =  —  V  F 


THE   CURVATURE   OF  SURFACES  413 


-/v 

The  dyadic  I  —  n  n  is  an  idemf  actor  for  all  vectors  perpen- 
dicular to  n  and  an  annihilator  for  vectors  parallel  to  n. 
Hence 

dn  •  (I  —  n  n)  =  d  n, 

and  V^.(I-nn)=0, 


Hence  dn  =  —  d  r  •  VV  F*  (I  —  nn). 

N 

But  dr  =  di  •  (I  —  nn). 

(I-nn)  •  VV.F.  (I-nn) 
Hence       dn  =  dr<- — - 

N 

Let  tfi  =  (I-nn).'.- 


N 

Then  d  n  =  d  r  •  0. 

In  the  vicinity  of  any  point  upon  a  surface  the  variation  d  n  of 
the  unit  normal  is  a  linear  function  of  the  variation  of  the 
radius  vector  r. 

The  dyadic  0  is  self-conjugate.     For 

N  0  c  =  (I  -  n  n),  •  ( VV  F)c  •  (I  -  n  n)  0. 

Evidently  (I  -  n  n)c  =  (I  -  n  n)  and  by  (6)  Art.  147  V  V  F 
is  self-conjugate.  Hence  0C  is  equal  to  0.  When  applied  to 
a  vector  parallel  to  n,  the  dyadic  0  produces  zero.  It  is  there- 
fore planar  and  in  fact  uniplanar  because  self-conjugate.  The 
antecedents  and  the  consequents  lie  in  the  tangent  plane  to 


414  VECTOR  ANALYSIS 

the  surface.     It  is  possible  (Art.  116)  to  reduce   0  to  the 

form 

0  =  a  i'i'  +  b  j'j'  (5) 

where  i'  and  j'  are  two  perpendicular  unit  vectors  lying  in  the 
tangent  plane  and  a  and  b  are  positive  or  negative  scalars. 

dn  =  dr  •  (a  i'i'  +  b  j'j'). 

The  vectors  i',  j'  and  the  scalars  a,  b  vary  from  point  to  point 
of  the  surface.     The  dyadic  0  is  variable. 

151.]  The  conic  r  •  0  •  r  =  1  is  called  the  indicatrix  of  the 
surface  at  the  point  in  question.  If  this  conic  is  an  ellipse, 
that  is,  if  a  and  b  have  the  same  sign,  the  surface  is  convex  at 
the  point ;  but  if  the  conic  is  an  hyperbola,  that  is,  if  a  and  b 
have  opposite  signs  the  surface  is  concavo-convex.  The  curve 
r .  0 .  r  —  1  may  be  regarded  as  approximately  equal  to  the 
intersection  of  the  surface  with  a  plane  drawn  parallel  to  the 
tangent  plane  and  near  to  it.  If  r  •  0  •  r  be  set  equal  to  zero 
the  result  is  a  pair  of  straight  lines.  These  are  the  asymp- 
totes of  the  conic.  If  they  are  real  the  conic  is  an  hyperbola ; 
if  imaginary,  an  ellipse.  Two  directions  on  the  surface  which 
are  parallel  to  conjugate  diameters  of  the  conic  are  called  con- 
jugate directions.  The  directions  on  the  surface  which  coin- 
cide with  the  directions  of  the  principal  axes  i',  j'  of  the 
indicatrix  are  known  as  the  principal  directions.  They  are  a 
special  case  of  conjugate  directions.  The  directions  upon  the 
surface  which  coincide  with  the  directions  of  the  asymptotes 
of  the  indicatrix  are  known  as  asymptotic  directions.  In  case 
the  surface  is  convex,  the  indicatrix  is  an  ellipse  and  the 
asymptotic  directions  are  imaginary. 

In  special  cases  the  dyadic  0  may  be  such  that  the  coeffi- 
cients a  and  b  are  equal.  0  may  then  be  reduced  to  the 
form 

0  =  a(i'i'+j'j')  (5)' 


THE   CURVATURE  OF  SURFACES  415 

in  an  infinite  number  of  ways.  The  directions  i'  and  j'  may  be 
any  two  perpendicular  directions.  The  indicatrix  becomes  a 
circle.  Any  pair  of  perpendicular  diameters  of  this  circle 
give  principal  directions  upon  the  surface.  Such  a  point  is 
called  an  umbilic.  The  surface  in  the  neighborhood  of  an 
umbilic  is  convex.  The  asymptotic  directions  are  imaginary. 
In  another  special  case  the  dyadic  0  becomes  linear  and  redu- 
cible to  the  form  &  =  a  i'i'.  (5)" 

The  indicatrix  consists  of  a  pair  of  parallel  lines  perpendicular 
to  i'.  Such  a  point  is  called  a  parabolic  point  of  the  surface. 
The  further  discussion  of  these  and  other  special  cases  will  be 
omitted. 

The  quadric  surfaces  afford  examples  of  the  various  kinds 
of  points.  The  ellipsoid  and  the  hyperboloid  of  two  sheets 
are  convex.  The  indicatrix  of  points  upon  them  is  an  ellipse. 
The  hyperboloid  of  one  sheet  is  concavo-convex.  The  in- 
dicatrix of  points  upon  it  is  an  hyperbola.  The  indicatrix 
of  any  point  upon  a  sphere  is  a  circle.  The  points  are  all 
umbilics.  The  indicatrix  of  any  point  upon  a  cone  or  cylinder 
is  a  pair  of  parallel  lines.  The  points  are  parabolic.  A  sur- 
face in  general  may  have  upon  it  points  of  all  types — elliptic, 
hyperbolic,  parabolic,  and  umbilical. 

152.]  A  line  of  principal  curvature  upon  a  surface  is  a 
curve  which  has  at  each  point  the  direction  of  one  of  the  prin- 
cipal axes  of  the  indicatrix.  The  direction  of  the  curve  at  a 
point  is  always  one  of  the  principal  directions  on  the  surface  at 
that  point.  Through  any  given  point  upon  a  surface  two  per- 
pendicular lines  of  principal  curvature  pass.  Thus  the  lines 
of  curvature  divide  the  surface  into  a  system  of  infinitesi- 
mal rectangles.  An  asymptotic  line  upon  a  surface  is  a  curve 
which  has  at  each  point  the  direction  of  the  asymptotes  of  the 
indicatrix.  The  direction  of  the  curve  at  a  point  is  always 
one  of  the  asymptotic  directions  upon  the  surface.  Through 


416  VECTOR  ANALYSIS 

any  given  point  of  a  surface  two  asymptotic  lines  pass.  These 
lines  are  imaginary  if  the  surface  is  convex.  Even  when  real 
they  do  not  in  general  intersect  at  right  angles.  The  angle 
between  the  two  asymptotic  lines  at  any  point  is  bisected  by 
the  lines  of  curvature  which  pass  through  that  point. 

The  necessary  and  sufficient  condition  that  a  curve  upon  a 
surface  be  a  line  of  principal  curvature  is  that  as  one  advances 
along  that  curve,  the  increment  of  d  n,  the  unit  normal  to  the 
surface  is  parallel  to  the  line  of  advance.  For 

dn=  &•  dr  =  (a  i'i'  +  I  j'j')  .  dr 
dr  =  x  i'  +  yj'. 

Then  evidently  d  n  and  d  r  are  parallel  when  and  only  when 
dt  is  parallel  to  i'  or  j7.  The  statement  is  therefore  proved. 
It  is  frequently  taken  as  the  definition  of  lines  of  curvature. 
The  differential  equation  of  a  line  of  curvature  is 

dnxdr  =  Q.  (6) 

Another  method  of  statement  is  that  the  normal  to  the  surface, 
the  increment  d  n  of  the  normal,  and  the  element  d  r  of  the 
surface  lie  in  one  plane  when  and  only  when  the  element  d  r 
is  an  element  of  a  line  of  principal  curvature.  The  differential 
equation  then  becomes 

[n    dn    dr]=0.  (7) 

The  necessary  and  sufficient  condition  that  a  curve  upon  a 
surface  be  an  asymptotic  line,  is  that  as  one  advances  along 
that  curve  the  increment  of  the  unit  normal  to  the  surface  is 
perpendicular  to  the  line  of  advance.  For 

dn  =  dr  •  0 
dn  •  dt  =  dr  •  0  •  dr. 

If  then  d  n  •  d  r  is  zero  d  r  •  0  •  d  r  is  zero.  Hence  d  r  is  an 
asymptotic  direction.  The  statement  is  therefore  proved.  It 


THE   CURVATURE  OF  SURFACES  417 

is  frequently  taken  as  the  definition  of  asymptotic  lines.     The 
differential  equation  of  an  asymptotic  line  is 

dvdr  =  Q.  (8) 

153.]  Let  P  be  a  given  point  upon  a  surface  and  n  the 
normal  to  the  surface  at  P.  Pass  a  plane  p  through  n.  This 
plane  p  is  normal  to  the  surface  and  cuts  out  a  plane  section. 
Consider  the  curvature  of  this  plane  section  at  the  point  P. 
Let  n'  be  normal  to  the  plane  section  in  the  plane  of  the 
section,  n'  coincides  with  n  at  the  point  P.  But  unless  the 
plane  p  cuts  the  surface  everywhere  orthogonally,  the  normal 
n'  to  the  plane  section  and  the  normal  n  to  the  surface  will  not 
coincide,  d  n  and  d  n'  will  also  be  different.  The  curvature 
of  the  plane  section  lying  in  p  is  (Art.  57). 


ds      d  s2 

As  far  as  numerical  value  is  concerned  the  increment  of  the 
unit  tangent  t  and  the  increment  of  the  unit  normal  n'  are 
equal.  Moreover,  the  quotient  of  d  r  by  d  s  is  a  unit  vector 
in  the  direction  of  d  n'.  Consequently  the  scalar  value  of  C  is 

dn'    dr  _dn'  •  dr 
ds     d  s         ds2 

By  hypothesis      n  =  n'  at  P    and    n  •  d  r  =  n'  •  d  r  =  0.; 
d(n»dr~)=dn'dr  +  n'dzr  =  (), 
d  (V  •  d  r)  =  d  n'  •  d  r  +  n'  •  d2  r  =  0. 

Hence        d  n  •  d  r  +  n  -  d2  r  =  d  n'  •  d  r  +  n'  •  d2  r. 

Since  n  and  n'  are  equal  at  P, 

dn  •  dr  =  dn'  •  dr. 

dn»  dr      dr  •  4>  •  dr      dr  •  <P  •  dr 
Hence      C  =    -  -^5-  rfr.rfr 

27 


418  VECTOR  ANALYSIS 


n  ,      I 

(J  •=.  d  —      -  •  -f-    0 

dr  •  dr         dr  •  dr 

Hence        C  =  a  cos2  (i',  d  r)  +  &  cos2  (j',  d  r), 
or  C  =  a  cos2  (i',  dr)  +  J  sin2  (i',  df).          (10) 

The  interpretation  of  this  formula  for  the  curvature  of  a 
normal  section  is  as  follows  :  When  the  plane  p  turns  about 
the  normal  to  the  surface  from  i'  to  j',  the  curvature  C  of  the 
plane  section  varies  from  the  value  a  when  the  plane  passes 
through  the  principal  direction  i',  to  the  value  &  when  it 
passes  through  the  other  principal  direction  j  '.  The  values 
of  the  curvature  have  algebraically  a  maximum  and  njinimum 
in  the  directions  of  the  principal  lines  of  curvature.  If  a  and 
6  have  unlike  signs,  that  is,  if  the  surface  is  concavo-convex 
at  P,  there  exist  two  directions  for  which  the  curvature  of  a 
normal  section  vanishes.  These  are  the  asymptotic  directions. 

154.]  The  sum  of  the  curvatures  in  two  normal  sections 
at  right  angles  to  one  another  is  constant  and  independent  of 
the  actual  position  of  those  sections.  For  the  curvature  in 
one  section  is 

01  =  a  cos2  (i',  c?r)  +  5  sin2  (i',  dr), 
and  in  the  section  at  right  angles  to  this 

Cz  =  a  sin2  (i',  dr)  +  &  cos2  (i',  dr). 
Hence  C1+C2  =  a  +  b=0s  (11) 

which  proves  the  statement. 

It  is  easy  to  show  that  the  invariant  02Sis  equal  to  the  pro- 
duct of  the  curvatures  a  and  b  of  the  lines  of  principal  curv- 
ature. 

<Pzs=ab 

Hence  the  equation     xz  —  4>s  x  +  &zs  =  0  (12) 


THE   CURVATURE  OF  SURFACES  419 

is  the  quadratic  equation  which  determines  the  principal  curv- 
atures a  and  b  at  any  point  of  the  surface.  By  means  of  this 
equation  the  scalar  quantities  a  and  b  may  be  found  in  terms 
oiF(x,y,z). 

(I-nn)  •  VVJF.  (I-nn) 


VV^-2_ 

N 

(nn  •  VV F •  nn)5  =  (nn  •  nn  •  VV^T)5=  (nn 
Hence  ®a  = 


(nn«  VV^)5  =  nn:  VV^=n.  VVJ?7.  n. 


V.V.P     V.. 
Hence  0S  =  —^-  -—  (13) 


These  expressions  may  be  written  out  in  Cartesian  coordinates, 
but  they  are  extremely  long.  The  Cartesian  expressions  for 
<PZS  are  even  longer.  The  vector  expression  may  be  obtained 

as  follows: 

(I-nn)2.(VVJQ2.(I-nn)2 


(I  —  nn)2  =  nn. 
Hence 


155.]     Given  any  curve  upon  a  surface.     Let  t  be  a  unit 
tangent  to  the  curve,  n  a  unit  normal  to  the  surface  and  m  a 


420  VECTOR  ANALYSIS 

vector  defined  as  n  X  t.  The  three  vectors  n,  t,  m  constitute 
an  i,  j,  k  system.  The  vector  t  is  parallel  to  the  element  d  r. 
Hence  the  condition  for  a  line  of  curvature  becomes 

t  x  d  n  =  0.  (15) 

Hence  m  •  d  n  =  0 

d  (m  •  n)  =  0  =  m  •  dn  +  n  •  dm. 
Hence  n  •  d  m  =  0. 

Moreover  m  •  d  m  =  0. 

Hence  t  x  d  m  =  0,  (16) 

or  dmxdn  =  Q.  (16)' 

The  increments  of  m  and  of  n  and  of  r  are  all  parallel  in  case  of 
a  line  of  principal  curvature. 

A  geodetic  line  upon  a  surface  is  a  curve  whose  osculating 
plane  at  each  point  is  perpendicular  to  the  surface.  That  the 
geodetic  line  is  the  shortest  line  which  can  be  drawn  between 
two  points  upon  a  surface  may  be  seen  from  the  following 
considerations  of  mechanics.  Let  the  surface  be  smooth  and 
let  a  smooth  elastic  string  which  is  constrained  to  lie  in  the 
surface  be  stretched  between  any  two  points  of  it.  The  string 
acting  under  its  own  tensions  will  take  a  position  of  equili- 
brium along  the  shortest  curve  which  can  be  drawn  upon  the 
surface  between  the  two  given  points.  Inasmuch  as  the 
string  is  at  rest  upon  the  surface  the  normal  reactions  of  the 
surface  must  lie  in  the  osculating  plane  of  the  curve.  Hence 
that  plane  is  normal  to  the  surface  at  every  point  of  the  curve 
and  the  curve  itself  is  a  geodetic  line. 

The  vectors  t  and  d  t  lie  in  the  osculating  plane  and  deter- 
mine that  plane.  In  case  the  curve  is  a  geodetic,  the  normal 
to  the  osculating  plane  lies  in  the  surface  and  consequently  is 
perpendicular  to  the  normal  n.  Hence 


THE   CURVATURE   OF  SURFACES  421 

n  •  t  X  d  t  =  0, 

n  X  t  •  d  t  =  0  (17) 

or  m  •  d  t  =  0. 

The  differential  equation  of  a  geodetic  line  is  therefore 

[n  d  r  d*  r]  =  0.  (18)  . 

Unlike  the  differential  equations  of  the  lines  of  curvature 
and  the  asymptotic  line,  this  equation  is  of  the  second  order. 
The  surface  is  therefore  covered  over  with  a  doubly  infinite 
system  of  geodetics.  Through  any  two  points  of  the  surface 
one  geodetic  may  be  drawn. 

As  one  advances  along  any  curve  upon  a  surface  there  is 
necessarily  some  turning  up  and  down,  that  is,  around  the 
axis  m,  due  to  the  fact  that  the  surface  is  curved.  There  may 
or  may  not  be  any  turning  to  the  right  or  left.  If  one  advances 
along  a  curve  such  that  there  is  no  turning  to  the  right  or 
left,  but  only  the  unavoidable  turning  up  and  down,  it  is  to  be 
expected  that  the  advance  is  along  the  shortest  possible  route 
—  that  is,  along  a  geodetic.  Such  is  in  fact  the  case.  The 
total  amount  of  deviation  from  a  straight  line  is  d  t.  Since  n, 
t,  m  form  an  i,  j,  k  system 

I  =  tt  +  nn  +  mm. 
Hence  dt  =  tt»dt  +  nn»dt  +  mm»dt. 

Since  t  is  a  unit  vector  the  first  term  vanishes.  The  second 
term  represents  the  amount  of  turning  up  and  down;  the 
third  term,  the  amount  to  the  right  or  left.  Hence  m  •  d  t  is 
the  proper  measure  of  this  part  of  the  deviation  from  a 
straightest  line.  In  case  the  curve  is  a  geodetic  this  term 
vanishes  as  was  expected. 

156.]  A  curve  or  surface  may  be  mapped  upon  a  unit 
sphere  by  the  method  of  parallel  normals.  A  fixed  origin  is 
assumed,  from  which  the  unit  normal  n  at  the  point  P  of  a 


422  VECTOR  ANALYSIS 

given  surface  is  laid  off.  The  terminus  P'  of  this  normal  lies 
upon  the  surface  of  a  sphere.  If  the  normals  to  a  surface  at  all 
points  P  of  a  curve  are  thus  constructed  from  the  same  origin, 
the  points  P'  will  trace  a  curve  upon  the  surface  of  a  unit 
sphere.  This  curve  is  called  the  spherical  image  of  the  given 
curve.  In  like  manner  a  whole  region  T  of  the  surface  may 
be  mapped  upon  a  region  T'  the  sphere.  The  region  T'  upon 
the  sphere  has  been  called  the  hodogram  of  the  region  T  upon 
the  surface.  If  d  r  be  an  element  of  arc  upon  the  surface  the 
corresponding  element  upon  the  unit  sphere  is 

d n  =  0  •  dr. 

If  da.  be  an  element  of  area  upon  the  surface,  the  corre- 
sponding element  upon  the  sphere  is  da!  where  (Art.  124). 

da!  =  (Pi*  da.. 
0  =  a  i'i'  +  J  j'j' 
02  =  ab  i'  x  j'  i'xj'  =  ab  nn. 
Hence  da!  =  ab  nn  •  da..  (19) 

The  ratio  of  an  element  of  surface  at  a  point  P  to  the  area  of 
its  hodogram  is  equal  to  the  product  of  the  principal  radii  of 
curvature  at  P  or  to  the  reciprocal  of  the  product  of  the  prin- 
cipal curvatures  at  P. 

It  was  seen  that  the  measure  of  turning  to  the  right  or  left 
is  m  •  d  t.  If  then  C  is  any  curve  drawn  upon  a  surface  the 
total  amount  of  turning  in  advancing  along  the  curve  is  the 
integral. 

(20) 


For  any  closed  curve  this  integral  may  be  evaluated  in  a 
manner  analogous  to  that  employed  (page  190)  in  the  proof 
of  Stokes's  theorem.  Consider  two  curves  C  and  C'  near 


THE   CURVATURE   OF  SURFACES  423 

together.     The  variation  which  the  integral  undergoes  when 
the  curve  of  integration  is  changed  from  C  to  C'  is 

8  I  m  •  d  t. 

B  Cm>dt=  Cs  (m.  dt~)=  C  Sm  •  dt+    Cm.Bdt 

d  (m  •  St)  =  dm*  St  +  m  •  d  Bt 
B  Cm-  dt=  C  Bm*dt-  C  dm  •  Bt  +   C  d  (m  -  5t). 

The  integral  of  the  perfect  differential  c£  (m  •  8  t)  vanishes 
when  taken  around  a  closed  curve.     Hence 

B  /  m»^t=  I  Sm  •  dt—  I  dm  •  Sk 
The  idemf  actor  is       I^tt  +  nn 


for  t  •  d  t  and  8  m  •  m  vanish.     A  similar  transformation  may 
be  effected  upon  the  term  d  m  •  St.     Then 

8  I  m«  dt=  /(Sm»n  n  •  dt  —  dm  •  n  n«8t). 

By  differentiating  the  relations  m  •  n  =  0  and  n  •  t  =  0  it  is 
seen  that 

Sm»n=:  —  m  •  Bn        n  •  Bt  =  —  8n«t 

dm  •  n  =  —  m  •  dn        n  •  dt  =  —  dn  •  t. 
Hence     S   \m*dt=\  (m»Sn  t>dn—  m  •  dn  t«Sn) 

B   Cm  •  dt=  |(mxt»Snxdn)=—  I  n  •  Bnx  dn. 


424  VECTOR  ANALYSIS 

The  differential  8n  x  dn  represents  the  element  of  area  in 
the  hodogram  upon  the  unit  sphere.     The  integral 


/  n  •  S  n  x  d  n  —   /  n  •  da! 


represents  the  total  area  of  the  hodogram  of  the  strip  of 
surface  which  lies  between  the  curves  0  and  C'.  Let  the 
curve  C  start  at  a  point  upon  the  surface  and  spread  out  to 
any  desired  size.  The  total  amount  of  turning  which  is  re- 
quired in  making  an  infinitesimal  circuit  about  the  point  is 
2  TT.  The  total  variation  hi  the  integral  is 

I   B  I  m  •  dt=  I  m  •  dt  —  2  TT. 


But  if  H  denote  the  total  area  of  the  hodogram. 
Hence  I  m  •  d  t  =  2  ?r  —  H9 

or  H=  2  TT  -   Cm  .  dt,  (22) 

or  H+  Cm  •  dt  =  2ir. 

The  area  of  the  hodogram  of  the  region  enclosed  by  any 
closed  curve  plus  the  total  amount  of  turning  along  that  curve 
is  equal  to  2  ?r.  If  the  surface  in  question  is  convex  the  area 
upon  the  sphere  will  appear  positive  when  the  curve  upon  the 
surface  is  so  described  that  the  enclosed  area  appears  positive. 
If,  however,  the  surface  is  concavo-convex  the  area  upon  the 
sphere  will  appear  negative.  This  matter  of  the  sign  of  the 
hodogram  must  be  taken  into  account  in  the  statement  made 
above. 


THE   CURVATURE   OF  SURFACES  425 

157.]  If  the  closed  curve  is  a  polygon  whose  sides  are 
geodetic  lines  the  amount  of  turning  along  each  side  is  zero. 
The  total  turning  is  therefore  equal  to  the  sum  of  the  exterior 
angles  of  the  polygon.  The  statement  becomes :  the  sum  of 
the  exterior  angles  of  a  geodetic  polygon  and  of  the  area  of 
the  hodogram  of  that  polygon  (taking  account  of  sign)  is 
equal  to  2  TT.  Suppose  that  the  polygon  reduces  to  a  triangle. 
If  the  surface  is  convex  the  area  of  the  hodogram  is  positive 
and  the  sum  of  the  exterior  angles  of  the  triangle  is  less  than 
2  TT.  The  sum  of  the  interior  angles  is  therefore  greater  than 
TT.  The  sphere  or  ellipsoid  is  an  example  of  such  a  surface. 
If  the  surface  is  concavo-convex  the  area  of  the  hodogram  is 
negative.  The  sum  of  the  ulterior  angles  of  a  triangle  is  in 
this  case  less  than  TT.  Such  a  surface  is  the  hyperboloid  of  one 
sheet  or  the  pseudosphere.  There  is  an  intermediate  case  in 
which  the  hodogram  of  any  geodetic  triangle  is  traced  twice  in 
opposite  directions  and  hence  the  total  area  is  zero.  The  sum 
of  the  interior  angles  of  a  triangle  upon  such  a  surface  is  equal 
to  TT.  Examples  of  this  surface  are  afforded  by  the  cylinder, 
cone,  and  plane. 

A  surface  is  said  to  be  developed  when  it  is  so  deformed  that 
lines  upon  the  surface  retain  their  length.  Geodetics  remain 
geodetics.  One  surface  is  said  to  be  developable  or  applicable 
upon  another  when  it  can  be  so  deformed  as  to  coincide  with 
the  other  without  altering  the  lengths  of  lines.  Geodetics 
upon  one  surface  are  changed  into  geodetics  upon  the  other. 
The  sum  of  the  angles  of  any  geodetic  triangle  remain  un- 
changed by  the  process  of  developing.  From  this  it  follows 
that  the  total  amount  of  turning  along  any  curve  or  the -area 
of  the  hodogram  of  any  portion  of  a  surface  are  also  invariant 
of  the  process  of  developing. 


426  VECTOR  ANALYSIS 


Harmonic  Vibrations  and  Bivectors 

158.]     The  differential  equation  of  rectilinear  harmonic 
motion  is 


.   „ 
dt* 

The  integral  of  this  equation  may  be  reduced  by  a  suitable 
choice  of  the  constants  to  the  form 

x  =  A  sin  n  t. 

This  represents  a  vibration  back  and  forth  along  the  JT-axis 
about  the  point  x  =  0.  Let  the  displacement  be  denoted  by 
D  in  place  of  x.  The  equation  may  be  written 

D  =  i  A  sin  n  t. 
Consider  D  =  i  A  sin  n  t  cos  m  x. 

This   is  a  displacement  not   merely  near  the  point  x  =  0 

2  k  TT 

but  along  the  entire  axis  of  x.    At  points  x  =  -  ,   where 

m 

Te  is  a  positive  or  negative  integer,  the  displacement  is  at  all 
times  equal  to  zero.  The  equation  represents  a  stationary 
wave  with  nodes  at  these  points.  At  points  midway  between 
these  the  wave  has  points  of  maximum  vibration.  If  the 
equation  be  regarded  as  in  three  variables  #,  y,  z  it  repre- 
sents a  plane  wave  the  plane  of  which  is  perpendicular  to 
the  axis  of  the  variable  x. 

The  displacement  given  by  the  equation 

Dj  =  i  A  !  cos  (m  x  —  n  f)  (1) 

is  likewise  a  plane  wave  perpendicular  to  the  axis  of  x  but 
not  stationary.  The  vibration  is  harmonic  and  advances 
along  the  direction  i  with  a  velocity  equal  to  the  quotient  of 


HARMONIC   VIBRATIONS  AND  BIVECTORS         427 

n  by  m.  If  v  be  the  velocity;  p  the  period;  and  I  the  wave 
length, 

n  2?r  2  TT  I 

v  =  -,        p  =  —  ,         /  =  -  ,        v  =  -.       (2) 
m  n  m  p 

The  displacement 

D2=j   Azcos(mx  —  nt) 

differs  from  DJ  in  the  particular  that  the  displacement  takes 
place  in  the  direction  j,  not  in  the  direction  i.  The  wave  as 
before  proceeds  in  the  direction  of  x  with  the  same  velocity. 
This  vibration  is  transverse  instead  of  longitudinal.  By  a 
simple  extension  it  is  seen  that 

D  =  A  cos  (m  x  —  n  f) 

is  a  displacement  in  the  direction  A.  The  wave  advances 
along  the  direction  of  x.  Hence  the  vibration  is  oblique  to 
the  wave-front.  A  still  more  general  form  may  be  obtained 
by  substituting  m  •  r  for  m  x.  Then 

D  =  A  cos  (m  •  r  —  n  t).  (3) 

This  is  a  displacement  in  the  direction  A.  The  maximum 
amount  of  that  displacement  is  the  magnitude  of  A.  The 
wave  advances  in  the  direction  m  oblique  to  the  displace- 
ment; the  velocity,  period,  and  wave-length  are  as  before. 
So  much  for  rectilinear  harmonic  motion.  Elliptic  har- 
monic motion  may  be  defined  by  the  equation  (p.  117). 


=  —  wr. 


o 
dt* 

The  general  integral  is  obtained  as 

r  =  A  cos  n  t  +  B  sin  n  t. 

The  discussion  of  waves  may  be  carried  through  as  pre- 
viously. The  general  wave  of  elliptic  harmonic  motion 
advancing  in  the  direction  m  is  seen  to  be 


428  VECTOR  ANALYSIS 

D  =  A  cos  (m  •  r  —  n  f)  —  B  sin  (m  •  r  —  n  t).         (4) 
dV 


dt 


=  —  7i  ]  A  sin  (m  •  r  —  n  t)  +  B  sin  (m  •  r  —  n  f)  \    (5) 


is  the  velocity  of  the  displaced  point  at  any  moment  in  the 
ellipse  in  which  it  vibrates.  This  is  of  course  entirely  differ- 
ent from  the  velocity  of  the  wave. 

An  interesting  result  is  obtained  by  subtracting  the  dis- 
placement from  the  velocity  multiplied  by  the  imaginary 
unit  V  —  1  and  divided  by  n. 


D- 


+  V  —  1  {  A  sin  (m  •  r  —  n  t)  +  B  sin  (m  •  r  —  n  t~)  }. 


The  expression  here  obtained,  as  far  as  its  form  is  concerned, 
is  an  imaginary  vector.  It  is  the  sum  of  two  real  vectors  of 
which  one  has  been  multiplied  by  the  imaginary  scalar  V  —  1- 
Such  a  vector  is  called  a  bivector  or  imaginary  vector.  The 
ordinary  imaginary  scalars  may  be  called  liscalars.  The  use 
of  bivectors  is  found  very  convenient  in  the  discussion  of 
elliptic  harmonic  motion.  Indeed  any  undamped  elliptic  har- 
monic plane  wave  may  be  represented  as  above  by  the  pro- 
duct of  a  bivector  and  an  exponential  factor.  The  real  part 
of  the  product  gives  the  displacement  of  any  point  and  the 
pure  imaginary  part  gives  the  velocity  of  displacement 
reversed  in  sign  and  divided  by  n. 

159.]  The  analytic  theory  of  bivectors  differs  from  that  of 
real  vectors  very  much  as  the  analytic  theory  of  biscalars 
differs  from  that  of  real  scalars.  It  is  unnecessary  to  have 
any  distinguishing  character  for  bivectors  just  as  it  is  need- 


HARMONIC  VIBRATIONS  AND  BIVECTORS        429 

less  to  have  a  distinguishing  notation  for  biscalars.  The  bi- 
vector may  be  regarded  as  a  natural  and  inevitable  extension 
of  the  real  vector.  It  is  the  formal  sum  of  two  real  vectors 
of  which  one  has  been  multiplied  by  the  imaginary  unit  V—  !• 
The  usual  symbol  i  will  be  maintained  for  V  —  1.  There  is 
not  much  likelihood  of  confusion  with  the  vector  i  for  the 
reason  that  the  two  could  hardly  be  used  in  the  same  place 
and  for  the  further  reason  that  the  Italic  i  and  the  Clarendon 
i  differ  considerably  in  appearance.  Whenever  it  becomes 
especially  convenient  to  have  a  separate  alphabet  for  bivec- 
tors  the  small  Greek  or  German  letters  may  be  called  upon. 
A  bivector  may  be  expressed  in  terms  of  i,  j,  k  with  com- 
plex coefficients. 

If  r  =  TJ  +  *  ra 

and  rl  =  xl  i  -f  yl  j  +  z^ 

r2  =  #2  *  +  2/2  J  +  *2  k> 

r  =  Oi  +  i  «2)  *  +  (Vi  +  *  y-i)  J  +  Oi  +  *  *a)  k> 
or  r  =  #i  +  yj  +  2k. 

Two  bivectors  are  equal  when  their  real  and  their  imaginary 
parts  are  equal.  Two  bivectors  are  parallel  when  one  is  the 
product  of  the  other  by  a  scalar  (real  or  imaginary).  If 
a  bivector  is  parallel  to  a  real  vector  it  is  said  to  have  a  real 
direction.  In  other  cases  it  has  a  complex  or  imaginary 
direction.  The  value  of  the  sum,  difference,  direct,  skew, 
and  indeterminate  products  of  two  bivectors  is  obvious  with- 
out special  definition.  These  statements  may  be  put  into 
analytic  form  as  follows. 

Let  r  =  TJ  +  i  r2     and    s  =  s2  +  i  sa. 

Then  if  r  =  s,        ri  =  »i  and  r2  =  s2 

if  r||s          r  =  a;s  =  (xl  +  i x^)  B, 


430  VECTOR  ANALYSIS 

r  +  s  =  (rx  +  BJ)  +  i  (r2  +  sa), 
r  .  B  =  (T!  •  «!  -  ra  •  sa)  +  *  (TJ  •  s2  +  r2  •  gj), 
r  x  s  =  (rx  X  B!  —  r2  x  s2)  +  i  (TI  x  s2  +  r2  x  Sj) 
rs  =  (rj  BJ  +  r2  Ba)  +  t  <>!  s2  +  r2  BJ). 

Two  bivectors  or  biscalars  are  said  to  be  conjugate  when 
their  real  parts  are  equal  and  their  pure  imaginary  parts 
differ  only  in  sign.  The  conjugate  of  a  real  scalar  or  vector 
is  equal  to  the  scalar  or  vector  itself.  The  conjugate  of  any 
sort  of  product  of  bivectors  and  biscalars  is  equal  to  the  pro- 
duct of  the  conjugates  taken  in  the  same  order.  A  similar 
statement  may  be  made  concerning  sums  and  differences. 

Oi  +  i  ra)  •  <>!  -  i  ra)  =  TJ  .  rx  +  ra  •  ra, 

(TJ  +  i  ra)  X  (rx  -  i  ra)  =  2  i  r2  x  rx, 
Oi  +  »  r2>  (ri  ~  *  r2)  =  (ri  ri  +  ra  r2)  +  *  02  ri  -  ri  r2>- 

If  the  bivector  r  =  TJ  +  i  ra  be  multiplied  by  a  root  of  unity 
or  cyclic  factor  as  it  is  frequently  called,  that  is,  by  an  imagi- 
nary scalar  of  the  form 

cos  q  +  i  sin  q  =  a  +  i  J,  (7) 

where  a2  +  &2  =  1, 

the  conjugate  is   multiplied  by  a  —  i  J,  and  hence  the  four 
products 

ri  '  ri  +  *2  *  r2>       *2  X  *H       Tl  r2  +  T2  r2»       r2  rl  ~  Tl  f2 

are  unaltered  by  multiplying  the  bivector  r  by  such  a  factor. 
Thus  if 

r'  =  r/  +  i  ra'  =  (a  +  i  5)  (rx  +  i  ra), 

ri' '  'i'  +  rz  '  'a'  =  TJ  •  rj  +  ra  •  ra,  etc. 


HARMONIC   VIBRATIONS  AND  BIVECTORS         431 

160.]  A  closer  examination  of  the  effect  of  multiplying  a 
bivector  by  a  cyclic  factor  yields  interesting  and  important 
geometric  results.  Let 

r/  +  i  r2'  =  (cos  q  +  i  sin  q)  (TJ  +  i  r2).         (8) 
Then  r/  —  rx  cos  q  —  r2  sin  #, 

ra'  =  ra  cos  1  +  ri  si*1  ?• 

By  reference  to  Art.  129  it  will  be  seen  that  the  change  pro- 
duced in  the  real  and  imaginary  vector  parts  of  a  bivector  by 
multiplication  with  a  cyclic  factor,  is  precisely  the  same  as 
would  be  produced  upon  those  vectors  by  a  cyclic  dyadic 

0  =  a  a'  +  cos  q  (bb'  +  c  c')  —  sin  q  (c  V  —  be') 

used  as  a  prefactor.  b  and  c  are  supposed  to  be  two  vectors 
collinear  respectively  with  rx  and  r2.  a  is  any  vector  not  in 
their  plane.  Consider  the  ellipse  of  which  rx  and  r2  are  a 
pair  of  conjugate  semi-diameters.  It  then  appears  that  r^ 
and  r2'  are  also  a  pair  of  conjugate  semi-diameters  of  that 
ellipse.  They  are  rotated  in  the  ellipse  from  TI  toward  r2,  by 
a  sector  of  which  the  area  is  to  the  area  of  the  whole  ellipse 
as  q  is  to  2  TT.  Such  a  change  of  position  has  been  called  an 
elliptic  rotation  through  the  sector  q. 

The  ellipse  of  which  TI  and  r2  are  a  pair  of  conjugate  semi- 
diameters  is  called  the  directional  ellipse  of  the  bivector  r. 
When  the  bivector  has  a  real  direction  the  directional  ellipse 
reduces  to  a  right  line  in  that  direction.  When  the  bivector 
has  a  complex  direction  the  ellipse  is  a  true  ellipse.  The 
angular  direction  from  the  real  part  TI  to  the  complex  part  r2 
is  considered  as  the  positive  direction  in  the  directional 
ellipse,  and  must  always  be  known.  If  the  real  and  imagi- 
nary parts  of  a  bivector  turn  in  the  positive  direction  in  the 
ellipse  they  are  said  to  be  advanced  ;  if  in  the  negative  direc- 
tion they  are  said  to  be  retarded.  Hence  multiplication  of  a 


432  VECTOR  ANALYSIS 

bivector  by  a  cyclic  factor  retards  it  in  its  directional  ellipse  by 
a  sector  equal  to  the  angle  of  the  cyclic  factor. 

It  is  always  possible  to  multiply  a  bivector  by  such  a  cyclic 
factor  that  the  real  and  imaginary  parts  become  coincident 
with  the  axes  of  the  ellipse  and  are  perpendicular. 

r  =  (cos  q  +  i  sin  q)  (a  +  i  b)  where  a  •  b  =  0. 
To  accomplish  the  reduction  proceed  as  follows  :  Form 
r  .  r  =  (cos  2  q  +  i  sin  2  q)  (a  +  i  b)  •  (a  +  i  b). 

If  a  •  b  =  0, 

r  •  r  =  (cos  2  q  +  i  sin  2  q)  (a  •  a  —  b  •  b). 

Let  r  •  r  =  a  +  i  b, 

and  tan  2  q  =  -. 

a 

With  this  value  of  q  the  axes  of  the  directional  ellipse  are 
given  by  the  equation 

a  +  i  b  =  (cos  q  —  i  sin  q)  r. 

In  case  the  real  and  imaginary  parts  a  and  b  of  a  bivector 
are  equal  in  magnitude  and  perpendicular  in  direction  both  a 
and  b  in  the  expression  for  r  •  r  vanish.  Hence  the  angle 
q  is  indeterminate.  The  directional  ellipse  is  a  circle.  A 
bivector  whose  directional  ellipse  is  a  circle  is  called  a  circu- 
lar bivector.  The  necessary  and  sufficient  condition  that  a 
non-vanishing  bivector  r  be  circular  is 

r  •  r  =  0,          r  circular. 
If  r  =  «i  +  2/j  +  2k, 

r  .  r  =  x2  +  y2  +  z2  =  0. 

The  condition  r  •  r  =  0,  which  for  real  vectors  implies  r  =  0, 
is  not  sufficient  to  ensure  the  vanishing  of  a  bivector.     The 


HARMONIC   VIBRATIONS  AND  BIVECTORS         433 

bivector  is  circular,  not  necessarily  zero.  The  condition  that 
a  bivector  vanish  is  that  the  direct  product  of  it  by  its  con- 
jugate vanishes. 

Ol  +  *  r2>   '    0*1  -  *  r2>  =  rl   '  rl  +  r2  '  f2  =  °> 

then  TI  =  r2  —  0  and  r  =  0. 

In  case  the  bivector  has  a  real  direction  it  becomes  equal  to 
its  conjugate  and  their  product  becomes  equal  to  r  •  r. 

161.]  The  condition  that  two  bivectors  be  parallel  is  that 
one  is  the  product  of  the  other  by  a  scalar  factor.  Any  bi- 
scalar  factor  may  be  expressed  as  the  product  of  a  cyclic 
factor  and  a  positive  scalar,  the  modulus  of  the  biscalar.  If 
two  bivectors  differ  by  only  a  cyclic  factor  their  directional 
ellipses  are  the  same.  Hence  two  parallel  vectors  have  their 
directional  ellipse  similar  and  similarly  placed — the  ratio  of 
similitude  being  the  modulus  of  the  biscalar.  It  is  evident 
that  any  two  circular  bivectors  whose  planes  coincide  are 
parallel.  A  circular  vector  and  a  non-circular  vector  cannot 
be  parallel. 

The  condition  that  two  bivectors  be  perpendicular 

is  r  •  6  =  0, 

or  TJ  •  sx  —  r2  •  s2  =  rx  •  s2  +  r2  •  BJ  =  0. 

Consider  first  the  case  in  which  the  planes  of  the  bivectors 
coincide.  Let 

r  =  a  (T!  +  i  r2),     s  =  b  (s1  +  i  s2). 

The  scalars  a  and  I  are  biscalars.  r1  may  be  chosen  perpen- 
dicular to  r2,  and  BI  may  be  taken  in  the  direction  of  ra.  The 
condition  r  •  s  =  0  then  gives 

r2  •  s2  =  0  and  TJ  •  s2  +.  r2  •  BJ  =  0. 

28 


434  VECTOR   ANALYSIS 

The  first  equation  shows  that  r2  and  s2  are  perpendicular  and 
hence  BI  and  s2  are  perpendicular.  Moreover,  the  second 
shows  that  the  angular  directions  from  i^  to  r2  and  from  sl  to 
s2  are  the  same,  and  that  the  axes  of  the  directional  ellipses 
of  r  and  s  are  proportional. 

Hence  the  conditions  for  perpendicularity  of  two  bivectors 
whose  planes  coincide  are  that  their  directional  ellipses  are 
similar,  the  angular  direction  in  both  is  the  same,  and  the 
major  axes  of  the  ellipses  are  perpendicular.1  If  both  vectors 
have  real  directions  the  conditions  degenerate  into  the  per- 
pendicularity of  those  directions.  The  conditions  therefore 
hold  for  real  as  well  as  for  imaginary  vectors. 

Let  r  and  s  be  two  perpendicular  bivectors  the  planes  of 
which  do  not  coincide.  Resolve  rx  and  r2  each  into  two  com- 
ponents respectively  parallel  and  perpendicular  to  the  plane 
of  s.  The  components  perpendicular  to  that  plane  contribute 
nothing  to  the  value  of  r  •  s.  Hence  the  components  of  rx 
and  r2  parallel  to  the  plane  of  s  form  a  bivector  r'  which  is 
perpendicular  to  s.  To  this  bivector  and  s  the  conditions 
stated  above  apply.  The  directional  ellipse  of  the  bivector  r' 
is  evidently  the  projection  of  the  directional  ellipse  of  r  upon 
the  plane  of  s. 

Hence,  if  two  bivectors  are  perpendicular  the  directional 
ellipse  of  either  bivector  and  the  directional  ellipse  of  the 
other  projected  upon  the  plane  of  that  one  are  similar,  have 
the  same  angular  direction,  and  have  their  major  axes  per- 
pendicular. 

162.]     Consider  a  bivector  of  the  type 

D  =  Ae,i(m'r-n()'  (9) 

where  A  and  m  are  bivectors  and  n  is  a  biscalar.  r  is  the 
position  vector  of  a  point  in  space.  It  is  therefore  to  be  con- 

1  It  should  be  noted  that  the  condition  of  perpendicularity  of  major  axes  is  not 
the  same  as  the  condition  of  perpendicularity  of  real  parts  and  imaginary  parts. 


HARMONIC   VIBRATIONS  AND  BIVECTORS        435 

sidered  as  real,     t  is  the  scalar  variable  time  and  is  also  to 
be  considered  as  real.     Let 


A  =  l 

^1  + 

*  A2, 

m  =  i 

ni  + 

im2, 

n  =  '. 

»!  + 

*wa, 

D  =  (A 

i  +  *  Aj) 

gidn, 

,.r  +  ,, 

D  =  (Ai 

+  *A.) 

g—  • 

"«-' 

As  has  been  seen  before,  the  factor  (Ax  +  i  A2)  e'(n>1  •*-"»" 
represents  a  train  of  plane  waves  of  elliptic  harmonic  vibra- 
tions. The  vibrations  take  place  in  the  plane  of  At  and  A2, 
in  an  ellipse  of  which  Ax  and  Aj  are  conjugate  semi-diam- 
eters. The  displacement  of  the  vibrating  point  from  the 
center  of  the  ellipse  is  given  by  the  real  part  of  the  factor. 
The  velocity  of  the  point  reversed  in  direction  and  divided 
by  Wj  is  given  by  the  pure  imaginary  part.  The  wave  ad- 
vances in  the  direction  nij.  The  other  factors  in  the  expres- 
sion are  dampers.  The  factor  e~ms*r  is  a  damper  in  the 
direction  m2.  As  the  wave  proceeds  in  the  direction  m2  it 
dies  away.  The  factor  entt  is  a  damper  in  time.  If  nz  is 
negative  the  wave  dies  away  as  time  goes  on.  If  ?22  is  posi- 
tive the  wave  increases  in  energy  as  time  increases.  The 
presence  (for  unlimited  time)  of  any  such  factor  in  an  ex- 
pression which  represents  an  actual  vibration  is  clearly  inad- 
missible. It  contradicts  the  law  of  conservation  of  energy. 
In  any  physical  vibration  of  a  conservative  system  nz  is  ne- 
cessarily negative  or  zero. 

The  general  expression  (9)  therefore  represents  a  train  of 
plane  waves  of  elliptic  harmonic  vibrations  damped  in  a 
definite  direction  and  in  time.  Two  such  waves  may  be  com- 
pounded by  adding  the  bivectors  which  represent  them.  If 
the  exponent  m  •  r  —  n  t  is  the  same  for  both  the  resulting 
train  of  waves  advances  in  the  same  direction  and  has  the 


430  VECTOR  ANALYSIS 

same  period  and  wave-length  as  the  individual  waves.  The 
vibrations,  however,  take  place  in  a  different  ellipse.  If  the 
waves  are 


*  (m  '  r  ~  " 


the  resultant  is  (  A  +  B)  e 


By  combining  two  trains  of  waves  which  advance  in  opposite 
directions  but  which  are  in  other  respects  equal  a  system  of 
stationary  waves  is  obtained. 


(—  mt  •  r  —  nt)   __ 

Ae-m*-re-<n'  (e<mi*r  +e-'mi-r)  =  2Acos  (n^  •  r)  e~^'T  e~int 

The  theory  of  bivectors  and  their  applications  will  not  be 
carried  further.  The  object  in  entering  at  all  upon  this  very 
short  and  condensed  discussion  of  bivectors  was  first  to  show 
the  reader  how  the  simple  idea  of  a  direction  has  to  give  way 
to  the  more  complicated  but  no  less  useful  idea  of  a  directional 
ellipse  when  the  generalization  from  real  to  imaginary  vectors 
is  made,  and  second  to  set  forth  the  manner  in  which  a  single 
bivector  D  may  be  employed  to  represent  a  train  of  plane 
waves  of  elliptic  harmonic  vibrations.  This  application  of  bi- 
vectors may  be  used  to  give  the  Theory  of  Light  a  wonderfully 
simple  and  elegant  treatment.1 

1  Such  use  of  bivectors  is  made  by  Professor  Gibbs  in  his  course  of  lectures  on 
"  The  Electromagnetic  Theory  of  Light,"  delivered  biannually  at  Yale  University. 
Bivectors  were  not  used  in  the  second  part  of  this  chapter,  hecause  in  the  opinion 
of  the  present  author  they  possess  no  essential  advantage  over  real  vectors  until 
the  more  advanced  parts  of  the  theory,  rotation  of  the  plane  of  polarization  by 
magnets  and  crystals,  total  and  metallic  reflection,  etc.,  are  reached, 


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